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4

Computational Approachesto Lattice Packing and Covering Problems

Achill Schurmann and Frank Vallentin∗

16th March 2004

Abstract

We describe algorithms which solve two classical problems in lattice geometry for any fixed dimen-sion d: the lattice covering and the simultaneous lattice packing–covering problem. Both algorithmsinvolve semidefinite programming and are based on Voronoi’sreduction theory for positive definitequadratic forms which describes all possible Delone triangulations ofZd. Our implementations verify allknown results in dimensionsd ≤ 5. Beyond that we attain complete lists of all locally optimalsolutionsfor d = 5. Ford = 6 our computations produce new best known covering as well as packing–coveringlattices which are closely related to the latticeE

∗

6.

AMS Mathematics Subject Classification 2000: 11H31

∗Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by theMinerva Foundation (Germany).

1

Contents

1 Overview 3

2 Basic Concepts and Notations 4

3 The Lattice Covering Problem 5

4 The Lattice Packing–Covering Problem 8

5 Voronoi’s Reduction Theory 95.1 Secondary Cones of Delone Triangulations . . . . . . . . . . . .. . . . . . . . . . . . . . 105.2 Bistellar Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 115.3 Main Theorem of Voronoi’s Reduction Theory . . . . . . . . . . .. . . . . . . . . . . . . . 125.4 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 135.5 Relations to the Theory of Secondary Polytopes . . . . . . . .. . . . . . . . . . . . . . . . 14

6 Convex Optimization with LMI Constraints 15

7 An LMI Constraint for the Inhomogeneous Minimum 16

8 Algorithms 188.1 Solving the Lattice Covering Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 188.2 Solving the Lattice Packing–Covering Problem . . . . . . . .. . . . . . . . . . . . . . . . 19

9 Properties of Local Optima 199.1 Convexity and Invariance of Sets of Local Optima . . . . . . .. . . . . . . . . . . . . . . . 209.2 Sufficient Conditions for Local Optima . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 219.3 Conditions for Isolated Local Optima . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 23

10 Local Lower Bounds via Moments of Inertia 23

11 Computational Results 2611.1 Dimensions 1, . . . , 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 2611.2 Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 2811.3 New Six–Dimensional Lattices . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 2811.4 Beautification and a Unified View . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 30

2

1 Overview

Classical problems in the geometry of numbers are the determination of most economical lattice spherepackings and coverings of the Euclideand–spaceEd. In this paper we describe algorithms for the latticecovering and the (simultaneous) lattice packing and covering problem.

Roughly speaking, both problems are concerned with the “most economical way” to coverEd. In thecase of the lattice covering problem the goal is to maximize the volume of a fundamental domain in a latticecovering with unit spheres. In other words, we want to minimize the number of unit spheres which areneeded to cover arbitrary large regions ofE

d. The objective of the lattice packing and covering problemis to maximize the minimal distance between lattice points in a lattice covering with unit spheres. Bothproblems have been solved for dimensionsd ≤ 5 only. The known results and their history are brieflyreviewed in Sections 3 and 4.

The intension of this paper is to collect and introduce to themathematical tools that allow us to solve thetwo problems computationally. Our implementations do not only verify all of the known results. We alsoattain additional information on all locally optimal solutions ford = 5, and we find new best known latticesfor both problems in dimension6. In particular we answer an open question by Ryshkov. In Section 11 wecollect all attained results.

The basic concepts and notations, which we use throughout the paper, are in Section 2. There we alsogive a precise definition of the two considered problems. Thereader familiar with sphere packings andcoverings, as well as lattices and their relation to positive definite quadratic forms, may skip this section.

Our algorithms and the known results by other authors are mainly based on a reduction theory for posi-tive definite quadratic forms by Voronoi. We give a detailed description of this main ingredient in Section 5with a special focus on computational implementability. The other main tool comes from convex opti-mization theory. Semidefinite programming problems and determinant maximization problems are brieflydescribed in Section 6. Both problems have in common that we want to minimize a convex function on vari-ables that satisfy some linear matrix inequalities (LMIs).In Section 7 we describe how the constraint that alattice gives aunit sphere covering can be modeled by LMIs. In Section 8 we mix theattained ingredientsand attain algorithms which solve the two considered problems.

Due to a combinatorial explosion our implementations of these algorithms only give complete solutionsfor d ≤ 5. Moreover, the used convex optimization algorithms are interior point methods and so theyyield only approximations. Therefore, in Section 9, we collect some mathematical tools which allow usto determine exact results from these approximations. In particular we can test computationally, whetheror not a given positive definite quadratic form gives a locally optimal solution. In the case of the latticepacking and covering problem we can test if such a solution isisolated. In order to run a heuristic searchfor good lattices it is necessary to have local lower bounds that we can compute fast. We describe one classof such bounds depending on the methods of inertia in Section10. We used them to find the new lattices indimension6.

3

2 Basic Concepts and Notations

To describe the situation more precisely, let us fix some notations. Unless stated otherwise, we refer thereader to [BR79], [GL87] and [CS88] for further informations about the introduced concepts.

Let Ed be ad–dimensional Euclidean space equipped with inner product〈·, ·〉, norm‖ · ‖ and unit ballBd =

{x ∈ E

d : ‖x‖ ≤ 1}

. A lattice L is a discrete subgroup inEd and from now on we will assumethat all latticesL have full rankd, that is, there exists a regular matrixA ∈ GLd(R) with L = AZd.The matrixA, respectively its columns, is called a basis ofL. All bases ofL are of the formAU withU ∈ GLd(Z). Thus, the determinantdet(L) = |det(A)| > 0 of the latticeL is well defined. We say thattwo latticesL andL′ belong to the same isometry class if for any basisA of L there is a basisA′ of L′ andan orthogonal transformationO ∈ Od(R) so thatA′ = OA.

The Minkowski sumL + αBd = {v + αx : v ∈ L,x ∈ Bd}, α ∈ R>0, is a lattice packing if thetranslates ofαBd have mutually disjoint interiors and a lattice covering ifE

d = L + αBd. The packingradiusλ(L) of a latticeL is given by

λ(L) = max{λ : L+ λBd is a lattice packing},

and the covering radiusµ(L) by

µ(L) = min{µ : L+ µBd is a lattice covering}.

The above values are attained: The packing radius is equal tohalf the length of a shortest non-zero vectorof L and the covering radius is equal to the maximum distance of points inEd to a closest lattice vector. Thepacking radius is the inradius of the Dirichlet–Voronoi polytope ofL

DV(L) = {x ∈ Ed : ‖x‖ ≤ ‖x− v‖,v ∈ L},

and the covering radius its circumradius. Both functionalsare homogeneous, that is, forα ∈ R we have

µ(αL) = |α|µ(L) and λ(αL) = |α|λ(L).

Thus, the covering density

Θ(L) =µ(L)d

det(L)· κd, κd = volBd,

is invariant with respect to scaling ofL. The same is true for the packing–covering constant

γ(L) =µ(L)

λ(L).

All these functionals are invariants of the isometry classes.In this paper we study the following two problems:

Problem 2.1. (Lattice Covering Problem)For a givend ∈ N, determineΘd = minLΘ(L), whereL ⊆ E

d runs over alld–dimensional lattices.

Problem 2.2. (Lattice Packing–Covering Problem)For a givend ∈ N, determineγd = minL γ(L), whereL ⊆ E

d runs over alld–dimensional lattices.

4

A brief description of history and results of both problems is given in Section 3 and Section 4. Theproblems were studied by using the intimate relation between lattices and positive definite quadratic forms(PQFs).

Let us describe this relation: To ad–dimensional latticeL = AZd with basisA we associate ad-dimensional PQF

Q[x] = xtAtAx = x

tGx,

where the Gram matrixG = AtA is symmetric and positive definite. We will carelessly identify quadraticforms with symmetric matrices by sayingQ = G andQ[x] = x

tQx. The set of quadratic forms is ad(d+1)2

dimensional real vector spaceSd, in which the set of PQFs forms an open, convex coneSd>0. Its closure isthe convex cone of all positive semi–definite quadratic formsSd≥0 which is pointed at0.

Note thatQ depends on the chosen basisA of L. Two arbitrary basesA andB of L are transformedinto each other by a unimodular transformation, that is, there exists anU ∈ GLd(Z) such thatA = BU .Thus,GLd(Z) acts onSd>0 by Q 7→ U tQU . Two PQFs lying in the same orbit under this action are calledarithmetically equivalent. This definition naturally extends to positive semi–definite quadratic forms.

Thus every lattice uniquely determines an arithmetical equivalence class of PQFs. On the other hand,every PQFQ admits a Cholesky decompositionQ = AtA, where the upper triangular matrixA is uniquelydetermined up to an orthogonal transformationO ∈ Od(R). Altogether, we have a bijection betweenisometry classes of latticesGLd(R)/Od(R) and arithmetical equivalence classes of PQFsSd>0/GLd(Z).

As a consequence, the lattice covering as well as the latticepacking–covering problem translate intoproblems for PQFs: The determinant (or discriminant) of a PQFQ is defined bydet(Q). The homogeneousminimumλ(Q) and the inhomogeneous minimumµ(Q) are given by

λ(Q) = minv∈Zd\{0}

Q[v], and µ(Q) = maxx∈Ed

minv∈Zd

Q[x− v].

A corresponding latticeL satisfiesdet(L) =√

det(Q), µ(L) =√

µ(Q), λ(L) =√

λ(Q)/2. Thereforeour goal is to minimize

Θ(Q) = Θ(L) =

õ(Q)d

detQ· κd, and γ(Q) = γ(L) = 2 ·

õ(Q)

λ(Q)

among all PQFsQ ∈ Sd>0.SinceΘ(Q) as well asγ(Q) are invariant with respect to the action ofGLd(Z) onSd>0, we only need to

consider one PQF in each arithmetical equivalence class. Finding a fundamental domain inSd>0 is one ofthe most basic and classical problems in geometry of numbers. Such areduction theoryfor PQFs, especiallysuitable for Problems 2.1 and 2.2 is due to Voronoi. We describe it in detail in Section 5.

3 The Lattice Covering Problem

The first who considered the problem was Kershner in 1939. In [Ker39] he showed that the hexagonal lattice(see Figure 1) gives the most economical sphere covering in the plane even without the restriction of beinga lattice covering.

5

Figure 1. The sphere covering given by the hexagonal lattice.

Since then the lattice covering problem has been solved up todimension5 (see Table 1). In all thesecases the latticeA∗

d, whose covering density equals

Θ(A∗d) =

√(d(d + 2)

12(d+ 1)

)d

(d+ 1) · κd,

provides the optimal lattice covering. Gameckii [Gam62], [Gam63], and Bleicher [Ble62] were the first whocomputed the covering density ofA∗

d for generald. They also showed that it is locally optimal with respectto covering density in every dimension.

The optimality of the body centered cubic latticeA∗3 whose Dirichlet-Voronoi polytope is a regular

truncated octahedron (the Dirichlet-Voronoi polytope ofA∗d is a regular permutahedron) was first proven by

Bambah [Bam54b]. Later, Barnes substantially simplified Bambah’s proof in [Bar56] and strengthened theresult by showing that in dimensions2 and3 the latticeA∗

d is the unique locally optimal lattice covering. Heused Voronoi’s reduction theory and anticipates that this is the right setup for solving the lattice coveringproblem in dimensions larger than three. Our algorithm in Section 8.1 confirms his anticipation. A thirdproof which is mainly elementary and unlike the previous twodoes not use any reduction theory of PQFswas given by Few [Few56]. At the moment no attempt is known to the authors to show that the optimalthree-dimensional lattice covering also gives an optimal sphere covering without lattice restriction.

In [Bam54a] Bambah conjectured that the latticeA∗4 gives the least dense four-dimensional lattice cov-

ering. In [DR63] Delone and Ryshkov proved Bambah’s conjecture. In [Bar65], [Bar66] Baranovskii gavean alternative proof of this fact. He determined all three locally optimal lattice coverings in dimension4.Dickson [Dic67] gave another alternative proof of this fact.

In a series of papers [Rys73], [BR73], [BR75], [RB75] Ryshkov and Baranovskii solved the latticecovering problem in dimension5. Due to space limitations the papers are very dense, so they prepared a140-pages long monograph [RB78] based on their investigations.

d lattice covering densityΘd

1 Z1 1

2 A∗2 1.209199

3 A∗3 1.463505

4 A∗4 1.765529

5 A∗5 2.124286

Table 1. Optimal lattice coverings.

6

In [Rys67] Ryshkov raised the question of finding the lowest dimensiond for which there is a betterlattice covering than the one given byA∗

d. In the same paper he showed thatA∗d is not the most efficient

lattice covering for all evend ≥ 114 and for all oddd ≥ 201. One of our main results is the answer toRyshkov’s question (see Theorem 11.3): Dimensiond = 6 is the lowest dimension for which there is abetter lattice covering than the one given byA∗

d. The proof is based on a computer search. In Section 8.1we will give an algorithm which finds all locally optimal lattice coverings in a given dimension. Using thisalgorithm we are able to verify all of the known results aboutoptimal lattice coverings up to dimension5.Unfortunately, due to a combinatorial explosion, the algorithm can not be applied practically in dimension6 or greater. Nevertheless we are able to find good lattice coverings in dimension6 by applying severalheuristics. We will give more details in Section 11.

d lattice densityΘ6 Lc16 2.4648037 A∗

7 3.0596218 A∗

8 3.6659499 A5

9 4.34018510 A∗

10 5.25171311 A4

11 5.59833812 A∗

12 7.51011313 A7

13 7.86406014 A5

14 9.00661015 A8

15 11.60162616 A∗

16 15.31092717 A∗

17 18.28781118 A∗

18 21.84094919 A∗

19 26.08182020 A∗

20 31.14344821 A∗

21 37.18456822 Λ∗

22 ≤ 27.883923 Λ∗

23 ≤ 15.321824 Λ24 7.903536

Table 2. Least dense known (lattice) coverings up to dimension24.

What else is known? In Table 2 we list all the least dense knownlattice coverings in dimensions6 to24. At the same time this list gives the least dense known spherecoverings: There is no covering of equalspheres known which is less dense than the best known latticecovering. This table is an update of Table2.1in [CS88].

It is not surprising that the Leech latticeΛ24 yields the best known lattice covering in dimension24. Thecovering density of the Leech lattice was computed by Conway, Parker and Sloane (Chapter 23 of [CS88]).It is not too brave to conjecture that the Leech lattice givesthe optimal24-dimensional sphere covering.Although we even do not know that the Leech lattice gives a locally optimal lattice covering. Using theLeech lattice Bambah and Sloane constructed in [BS82] a series of lattices in dimensionsd ≥ 24 whichgive a thinner lattice covering thanA∗

d. It seems that as a “corollary” of the existence of the Leech latticethe duals of the laminated latticesΛ22 andΛ23 give good lattice coverings. Their covering densities wereestimated by Smith [Smi88]. But we do not know the exact valuesΘ(Λ∗

22) andΘ(Λ∗23). For the definitions

of the root latticesAn,Dn,En, the laminated latticesΛn and much more we refer to [CS88].

7

In [Cox51] Coxeter gave a list of locally optimal latticepackings(extreme lattices) which are related toLie groups. One of them is the infinite series of locally densest packing latticesAr

d whered ≥ 2 andr > 1dividesd+1. The latticeAr

d is the unique sublattice ofA∗d containingAd to indexr. In [Bar94] Baranovskii

determined the covering density of the lattice covering given byA59 which is slightly better than the one

given byA∗9. Recently, Anzin extended Baranovskii’s work. In [Anz02] he computed the covering densities

of A411 andA7

13, and in a private communication he reported on computing thecovering densities ofA514 and

A815. They all give less dense lattice coverings than those provided by the correspondingA∗

d. We do notknow whether these lattice coverings are locally optimal. To answer Ryshkov’s question exhaustively it willbe necessary to further investigate lattice coverings in the dimensionsd = 7, 8, 10, 16, 17, . . . , 21.

4 The Lattice Packing–Covering Problem

The simultaneous lattice packing and covering problem, from now on in short “lattice packing–coveringproblem”, has been studied in different contexts and there are several different names and interpretationsof the lattice packing–covering constantγd. Lagarias and Pleasants [LP02] referred to it as the “Delonepacking–covering constant”. Ryshkov [Rys74] studied the equivalent problem of minimizing the density of(r,R)–systems. An(r,R)–system is a discrete point setX ⊆ R

d where (1) the distance between any twopoints ofX is at leastr and (2) the distance from any point inRd to a point inX is at mostR. If X is alattice, thenr = λ(X)/2 andR = µ(X).

Figure 2. A close sphere packing given by the hexagonal lattice.

Geometrically we may think of the lattice packing–coveringproblem as of maximizing the minimumdistance between lattice points in a lattice covering with unit spheres. Or, we could think of it as minimizingthe radius of a largest sphere that could additionally be packed into a lattice packing of unit spheres (seeFigure 2). Then, this minimal gap–radius is equal toγd − 1. Therefore the problem raised by L. Fejes Toth[Tot76] to find “close packings” attaining this gap–radiusis another formulation of the packing–coveringproblem.

The last interpretation shows thatγd ≥ 2 would imply that in anyd–dimensional lattice packing withspheres of unit radius there is still space for spheres of radius 1. In particular this would prove that indimensiond densest sphere packings are non–lattice packings. This phenomenon is likely to be true forlarge dimensions, but has not been verified for anyd so far.

Problem 4.1. Does there exist ad, such thatγd ≥ 2?

8

Note that this problem is in particular challenging in view of the asymptotic boundγd ≤ 2 + o(1) dueto Butler [But72].

As for the lattice covering problem, the lattice packing–covering has been solved up to dimension5 (seeTable 3). Ryshkov [Rys74] solved the general2–dimensional case. The3–dimensional case was settledby Boroczky [Bor86], even without the restriction to lattices. The4– and5–dimensional cases were bothsolved by Horvath [Hor82], [Hor86]. Note that the latticesHo4 andHo5 (see Section 11) discovered byHorvath are neither best covering, nor best packing lattices. As in the case of lattice coverings, the resultswere attained by using Voronoi’s reduction theory.

d lattice lattice packing–covering constantγd1 Z

1 12 A∗

22√3≈ 1.154700

3 A∗3

√5/3 ≈ 1.290994

4 Ho4

√2√3(√3− 1) ≈ 1.362500

5 Ho5

√3/2 +

√13/6 ≈ 1.449456

Table 3. Optimal packing–covering lattices.

Our computations described in Section 11 verify all of the known results in dimension≤ 5. As for thecovering problem, none of the valuesγd have been determined in a dimensiond ≥ 6 so far. In Section 11 wereport on a new best known packing–covering lattice ford = 6. We thereby in particular showγ6 < 1.42,revealing the phenomenonγ6 < γ5, recently suspected by Lagarias and Pleasants [LP02], and described byZong [Zon02] (Remark 3).

We were not able to find any new best known lattices in dimensions≥ 7 so far, so that e.g. the latticeE∗7 still gives the best known lattice in dimension7. Nevertheless, because of their symmetry and the known

bounds onγd, Zong [Zon02] made the following, very reasonable conjecture: E8 and Leech latticeΛ24 areoptimal in their dimensions. The corresponding known bounds are shown in Table 4. In dimension7 andbetween dimensions8 and24 we do not know enough so far to state any serious conjectures.The exactvalue for the smallest known lattice packing–covering constant in dimension6 is a lengthy expression thatwe will give in Section 11.

d lattice lattice packing–covering constantγd6 Lpc

6 ≈ 1.411081

7 E∗7

√7/3 ≈ 1.527525

8 E8

√2 ≈ 1.414213

24 Λ24

√2 ≈ 1.414213

Table 4. Lattice packings–covering records.

5 Voronoi’s Reduction Theory

The general task of a reduction theory for PQFs is to give a fundamental domain forSd>0/GLd(Z). This is asubset which behaves likeSd>0/GLd(Z) up to boundary identifications. There are many different reductiontheories, connected with names like Lagrange, Gauß, Hermite, Korkine, Zolotareff, Minkowski, Voronoi,and others (cf. [SO85]). In this section we describe Voronoi’s reduction theory (cf. [Vor08]) which is basedon Delone triangulations.

9

5.1 Secondary Cones of Delone Triangulations

Let Q ∈ Sd>0 be a PQF. A polytopeL = conv{v1, . . . ,vn}, with v1, . . . ,vn in Zd, is called aDelone

polytopeof Q if there exists ac ∈ Rd and a real numberr ∈ R with Q[vi − c] = r2 for all i = 1, . . . , n,

and for all other lattice pointsv ∈ Zd\{v1, . . . ,vn} we have strict inequalityQ[v − c] > r2. The set of all

Delone polytopesDel(Q) = {L : L is a Delone polytope ofQ}

is called theDelone subdivision(or L–partition) of Q. A Delone triangulationis a Delone subdivision thatconsists of simplices only. Note that we use the letterL to denote Delone polytopes for historical reasons(cf. [Vor08] and [Del38]).

The Delone subdivision of a PQF is a periodic polytopal subdivision of Rd. We say that two DelonepolytopesL andL′ areequivalentif there is av ∈ Z

d with L′ = L+ v. Given a Delone subdivisionD ofRd, the set of PQFs with Delone subdivisionD forms thesecondary cone

∆(D) := {Q ∈ Sd>0 : Del(Q) = D}.

Especially in the Russian literature it is often referred toas theL–type domainof Q ∈∆(D).Let Q be a PQF whose Delone subdivision is a triangulation ofR

d. In the following we will describethe secondary cone ofDel(Q). For this, letL = conv{v1, . . . ,vd+1} andL′ = conv{v2, . . . ,vd+2}be twod–dimensional Delone simplices ofQ sharing the common facetF = conv{v2, . . . ,vd+1}. Letα1, . . . , αd+2 be real numbers withα1 = 1,

∑d+2i=1 αi = 0 and

∑d+2i=1 αivi = 0 (henceαd+2 > 0). The

regulatorof the pair of adjacent simplices(L,L′) is the linear form (L,L′)(Q′) =

∑d+2i=1 αiQ

′[vi], Q′ ∈ Sd.In particular, note that the regulator solely depends on thepointsv1, . . . ,vd+2, that(L,L′)(Q) > 0, andthat (L,L′) = (L+v,L′+v) for all v ∈ Z

d. Due to the following Theorem (cf. [Vor08]) one can describe∆(Del(Q)) by linear inequalities coming from the (finitely many) regulators ofDel(Q).

Theorem 5.1. Let Q be a PQF whose Delone subdivision is a triangulation. The secondary cone of theDelone triangulationDel(Q) is the full–dimensional open polyhedral cone

∆(Del(Q)) = {Q′ ∈ Sd : (L,L′)(Q′) > 0, (L,L′) pair of adjacent simplices ofDel(Q)}.

Example 5.2. As a first example and because of its importance for the lattice problems introduced in Sec-tion 2, we describe the Delone subdivision ofVoronoi’s principal form of the first typeQ[x] = d

∑x2i −∑

xixj which is associated to the latticeA∗d in greater detail. The Delone subdivision ofQ is a trian-

gulation and can be described as follows: Lete1, . . . ,ed be the standard basis vectors ofZd, and set

ed+1 = −e1 − · · · − ed. For a permutationπ ∈ Sd+1 we define thed–dimensional simplexLπ by

Lπ = conv{eπ(1),eπ(1) + eπ(2), . . . ,eπ(1) + · · · + eπ(d+1)}.

The set of simplices{Lπ + v : v ∈ Zd, π ∈ Sd+1} defines a triangulation ofRd which we from now on

denote byD1. The full–dimensional cells of the star of the origin areLπ, π ∈ Sd+1. In the star two simplicesLπ andLπ′ have a facet in common if and only ifπ andπ′ differ by a single transposition of two adjacentpositions. The automorphism group ofD1 is isomorphic to the permutation groupSd+1. In Figure 3 the starof the origin in dimension2 is illustrated.

The secondary cone ofD1 is (cf. [Vor08], §102–104)

∆(D1) = {Q ∈ Sd : qij < 0, i 6= j, and∑

i qij > 0, j = 1, . . . , d }.

Its topological closure∆(D1) is calledVoronoi’s principal domain of the first type.

10

✻

✲r

r r

r

r

r

r

������

������������

������

Lid

L(23)

L(12)

L(13)

L(123)

L(132)

Figure 3. The triangulationD1 in dimensiond = 2.

It was shown by Voronoi that the topological closure of the secondary cones gives a facet–to–facettesselation ofSd≥0. By a Theorem of Gruber and Ryshkov [GR89] we even have a face–to–face tessellationbecause “facet–to–facet implies face–to–face”.

5.2 Bistellar Operations

Now, given a secondary cone, how do we find its neighbours, that is, those secondary cones sharing a facetwith the given one?

An answer can be given by taking a closer look onto the so–called repartitioning polytopes(introducedby Ryshkov and Baranovskii in [RB78]), which are “hidden” inthe definition of the regulators. Repartition-ing polytopes ared–dimensional Delone polytopes having a representation as the convex hull of two Delonesimplices sharing a common facet. Thus, repartitioning polytopes haved+ 2 vertices.

Generally,d–dimensional polytopes withd + 2 vertices have the special property that there are ex-actly two different ways to triangulate them: LetV be a set ofd + 2 points which affinely spansRd. Let∑

v∈V αvv = 0,∑

v∈V αv = 0, be an affine relation between these points. There exist exactly two trian-gulations ofconv V : T+(V,α) with d–simplicesconv(V \{v}), αv > 0, andT−(V,α) with d–simplicesconv(V \{v}), αv < 0 (cf. e.g. [GKZ94]).

LetD be a Delone triangulation ofRd and letF be a(d−1)–dimensional cell ofD. Then,F is containedin two simplicesL andL′ of D. By V we denote the set of vertices ofL andL′, V = vertL ∪ vertL′. Byα we denote an affine relation between the points inV . The(d−1)–dimensional cellF is called aflippablefacetof the triangulationD if one of the triangulationsT+(V,α) or T−(V,α) is a subcomplex ofD. If Fis a flippable facet ofD and we replace the subcomplexT+(V,α) by T−(V,α) [respectivelyT−(V,α) byT+(V,α)], then we get a new triangulation. This replacement is called bistellar operationor flip.

Notice that non–flippable facets do exist and that performing a bistellar operation in a Delone triangula-tion does not necessarily produce a Delone triangulation. Both phenomena occur starting from dimension4.

Nevertheless, the facets of∆(D) give exactly those bistellar operations ofD which yield new Delonetriangulations. A(d − 1)–dimensional cellL ∩ L′ ∈ D is a flippable facet whenever the correspondingregulator(L,L′) gives a facet–defining hyperplane of∆(D) (see [Vor08],§87–§88). This is clear since therepartitioning polytopeconv(L ∪ L) is a Delone polytope of the PQFs lying in the relative interior of thefacet given by (L,L′).

LetF be a facet of∆(D). We describe how the Delone triangulationD changes if we vary a PQF contin-

11

Figure 4. The graph of two-dimensional Delone triangulations.

uously: We start from the interior of∆(D), then we move towards a relative interior point ofF and finallywe go infinitesimally further, leaving∆(D). In every repartitioning polytopeV = conv(L ∪ L′) whereL,L′ is a pair of adjacent simplices whose regulator definesF, i.e. linF = {Q ∈ Sd : (L,L′)(Q) = 0}, weperform a bistellar operation. This gives a new triangulationD′, which is again a Delone triangulation. Thetwo secondary cones∆(D) and∆(D′) have the complete facetF in common. We say thatD andD′ arebistellar neighbours. In [Vor08] §91–§96, Voronoi computes the secondary cone ofD′ explicitly and showsthat∆(D′) has dimensiond(d+1)

2 .

5.3 Main Theorem of Voronoi’s Reduction Theory

By constructing bistellar neighbours we could produce infinitely many Delone triangulations starting fromthe Delone triangulationD1 of Voronoi’s principal form of the first type (a part of the infinite flip graphoftwo–dimensional Delone triangulations is given in Figure 4). But many of these will not be essentially new,because the groupGLd(Z) is acting on the set of Delone subdivisions by(A,D) 7→ AD and it is acting onthe set of secondary cones by(A,∆) 7→ At

∆A, A ∈ GLd(Z). We are only interested in the orbits of thesegroup actions and there are only finitely many, as shown by Voronoi (cf. also Deza, Grishukhin and Laurentin [DL97], Chapter 13.3). Altogether we get:

Theorem 5.3. (Main Theorem of Voronoi’s Reduction Theory)The topological closures of secondary cones of Delone triangulations give a face–to–face tiling ofSd≥0. Twosecondary cones share a facet if and only if they are bistellar neighbours. The groupGLd(Z) acts on thetiling, and under this group action there are only finitely many non–equivalent secondary cones.

The main Theorem translates into Algorithm 1 which enumerates all non–equivalent Delone triangula-tions in a given dimension.

We implemented Algorithm 1 and succeeded to classify all non–equivalent Delone triangulations up todimension5. The classification of all non–equivalent Delone triangulations in dimension6 seems to be atthe limit of what is computational possible at the moment andwe were not able to finish it until now. Wefound more than250, 000 non–equivalent triangulations and we think that there are several millions. Table 5shows the numbers.

We are not the first who got this classification. Voronoi performed the classification of all non–equivalentDelone triangulation up to dimension4 in his memoir [Vor08]. In [BR73] and [RB78] Ryshkov and Bara-novskii reported on221 non–equivalent Delone triangulation in dimension5. But they missed one type

12

Input: Dimensiond.Output: SetR of all non–equivalentd–dimensional Delone triangulations.

T ← {D1}. R← ∅.while there is aD ∈ T do

T ← T\{D}. R← R∪ {D}.compute the regulators ofD.compute the facetsF1, . . . , Fn of ∆(D).for i = 1, . . . , n do

compute the bistellar neighbourDi of D which is defined byFi.if Di is not equivalent to a Delone triangulation inR∪ {D1, . . . ,Di−1} thenT ← T ∪ {Di}.

end ifend for

end whileAlgorithm 1. Enumeration of all non–equivalent Delone triangulations.

which was found by Engel [Eng98]. Grishukhin and Engel [EG02] undertook the non–trivial task to iden-tify the Delone triangulation missing in the list of Ryshkovand Baranovskii. There they also report onseveral errors in both lists. Our computations confirm the number of222 Delone triangulations in dimen-sion5. Beginning with dimension6 the number of non–equivalent Delone triangulations startsto explodeand not much is known about the number of non–equivalent Delone triangulations for general dimensiond.

Dimension 1 2 3 4 5 6

# Delone triangulations 1 1 1 3 222 > 250, 000

Table 5. Numbers of non–equivalent Delone triangulations.

5.4 Degeneracy

Until now we have only dealt with Delonetriangulationsof positivedefinitequadratic forms. Let us lookat possible degenerations — Delone subdivisions of positive semi–definite quadratic forms — and find outhow they fit into the theory developed so far.

Let Q be a positive semi–definite quadratic form which is arithmetically equivalent to( 0 00 Q′

)whereQ′

is positive definite. Then, we can use the definition of Delonesubdivision almost literally, we only have toreplace “polytope” by “polyhedron”.

Delone subdivisions are limiting cases of triangulations.Their secondary cones occur on the boundariesof full–dimensional secondary cones of Delone triangulations. LetD andD′ be two Delone subdivisions.We sayD is arefinementofD′ if every Delone polytope ofD is a subset of some Delone polytope ofD′. Thefollowing Proposition which seems to be folklore shows thatthe relation between refinements, secondarycones and sums of positive semi–definite quadratic forms is very natural. One can find a proof e.g. in[Val03].

Proposition 5.4. LetD be a Delone triangulation.

1. A positive semi–definite quadratic formQ lies in∆(D) if and only ifD is a refinement ofDel(Q).

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2. If two positive semi–definite quadratic formsQ andQ′ both lie in∆(D), thenDel(Q + Q′) is acommon refinement ofDel(Q) andDel(Q′).

Therefore, the classification of all non–equivalent Delonesubdivisions is equivalent to the classificationof all non–equivalent secondary cones. This has been done upto dimension5. The1 and2–dimensionalcases are trivial, the3–dimensional case goes back to Federov who classified all polytopes which tile three-dimensional space by translates in 1885. Delone [Del29] (later corrected by Stogrin [Sto73]) found51 ofthe52 Delone subdivisions in dimension4. Recently, Engel [Eng00] reported that there are179, 372 non–equivalent five–dimensional Delone subdivisions. Again, the number in dimension6 is not known and willbe much larger. Table 6 summarizes this discussion.

Dimension 1 2 3 4 5 6

# Delone subdivisions 1 2 5 52 179, 372 ≫ 250, 000

Table 6. Numbers of non–equivalent Delone subdivisions.

5.5 Relations to the Theory of Secondary Polytopes

Triangulations of discrete point sets have attracted many researchers in recent years. They have manyapplications, e.g. in computational geometry, optimization, algebraic geometry, topology, etc. One maintool to understand the structural behavior of triangulations of finite point sets is the theory of secondarypolytopes invented by Gel’fand, Kapranov and Zelevinsky ([GKZ94]). We describe how this theory isrelated to Voronoi’s reduction theory, since despite different set–ups there are many similarities.

LetA = {a1, . . . ,an} ⊆ Rd be a finite set of points. Letw : A → R be a map that assigns to every

point in A a weight. The set of weight maps forms a vector space overR which we denote byRA. Alifting map l : A → R

d × R, l(ai) = (ai, w(ai)) is defined byw which lifts each pointai ∈ A on itsweightw(ai). A subdivision of the convex polytopeconvA is induced byl: We take the convex hull of thelifted pointsconv l(A) and project its lower faces as seen from(0,−∞) back down ontoRd. A subdivisionthat can be obtained in this manner is called aregular subdivision. Delone subdivisions (or more preciselyDelone subdivisions of finitely many points) are regular subdivisions since the underlying positive semi–definite quadratic form is the weight function. This view on Delone subdivisions was introduced by Brown[Bro79] and by Edelsbrunner and Seidel [ES86].

Let T be a regular triangulation ofconvA. We may ask what are the weight functions which defineT .What is thesecondary coneof T in the parameter spaceRA? Like in Voronoi’s reduction theory it turns outthat the secondary cone ofT is a full–dimensional open polyhedral cone. The topological closures of thesecondary cones of all regular triangulations tiles the spaceRA face–to–face. The tiling is calledsecondaryfan of A. If two secondary cones have a facet in common, then the corresponding regular triangulationsdiffer by a bistellar operation in exactly one “repartitioning polytope” (in this context it is a polytope withd + 2 vertices without the condition of being a Delone polytope) that is defined by the facet. The faces inthe secondary fanA are in a one–to–one correspondence to regular subdivisionsin the essentially the sameway we discussed in Section 5.4 for Delone subdivisions.

So far we have seen that the theory of regular subdivisions offinite point sets and the theory of Delonesubdivisions of the latticeZd can be developed analogously. But there are differences: The parameterspaces are of completely different nature. For regular subdivisions it is the vector spaceRA and for Delonesubdivisions we have the pointed coneSd≥0. Groups play an important role for Delone subdivisions. ThegroupZd is acting on Delone subdivisions by translations. On the setof secondary cones the groupGLd(Z)is acting.

14

If we order all regular subdivisions ofconvA by refinement we get a poset. This poset has a very nicecombinatorially structure as proved by Gel’fand, Kapranovand Zelevinsky: There exists a polytope — thesecondary polytopeΣ(A) ofA— whose normal fan equals the secondary fan ofA. So the refinement posetis anti–isomorphic to the face lattice of the secondary polytope. Regular triangulations are in one–to–onecorrespondence to the vertices, two regular triangulations differ by a bistellar operation if and only if theirvertices are connected by an edge, etc. We do not know if thereis a similar geometrical or combinatorialstructure lurking behind the refinement poset of Delone subdivisions.

Question 5.5.Does there exist something similar to the secondary polytope for Delone subdivisions?

6 Convex Optimization with LMI Constraints

In this section we introduce determinant maximization problems, which are convex programming problemswith linear matrix inequality constraints. In a sense they are equivalent to the more popular semidefinite pro-gramming problems. For both classes efficient algorithms and implementations are available. In Section 8we will see how Problem 2.1 and Problem 2.2 can naturally be formulated as a finite number of determinantmaximization problems.

Following Vandenberghe, Boyd, and Wu [VBW98] we say that adeterminant maximization problemisan optimization problem of the form

minimize ctx− log detG(x)

subject to G(x) is a positive definite matrix,F (x) is a positive semidefinite matrix.

The optimization vector isx ∈ Rd, the objective vector isc ∈ R

d, andG : Rd → Rm×m andF : Rd →

Rn×n are affine maps:

G(x) = G0 + x1G1 + · · ·+ xdGd,F (x) = F0 + x1F1 + · · ·+ xdFd,

whereGi ∈ Rm×m,Fi ∈ R

n×n, i = 0, . . . , d, are symmetric matrices. In the sequel we will writeG(x) ≻ 0andF (x) � 0 for the linear matrix inequalitiesdefining the constraints of the determinant maximizationproblem.

Note that we minimizectx − log detG(x) instead of maximizing the more intuitivectx + detG(x)because only in the first formulation the determinant maximization problem is a convex programming prob-lem. As a special case, our formulation reduces to a semidefinite programming problem, wheneverG(x) isthe identity matrix for allx ∈ R

d.Currently there exist two different types of algorithms — ellipsoid and interior–point methods — which

efficiently solve semidefinite programming problems. They can approximate the solution of a semidefiniteprogramming problem within any specified accuracy and run inpolynomial time if the instances are “well–behaving”. For more information on the exciting topic of semidefinite programming the interested reader isreferred to the vast amount of literature which to a great extend is available on the World Wide Web. A goodstarting point is the web site1 of Christoph Helmberg.

Nesterov and Nemirovskii [NN94] developed a framework for the design of efficient interior–pointalgorithms for general and specific classes of convex programming problems. There (§6.4.3), they also

1http://www-user.tu-chemnitz.de/˜helmberg/semidef.html

15

showed that the determinant maximization problem can be transformed into a semidefinite programmingproblem by a transformation which can be computed in polynomial time. Nevertheless it is faster to solvethe determinant maximization problem directly. Vandenberghe, Boyd, Wu [VBW98] and independently Toh[Toh99] gave interior–point algorithms for the determinant maximization problem. Both algorithms fit intothe general framework of Nesterov and Nemirovskii. For our implementation we use the software packageMAXDET2 of Wu, Vandenberghe, and Boyd as a subroutine.

7 An LMI Constraint for the Inhomogeneous Minimum

In this section we will give a linear matrix inequality in theparameters(qij) of a PQFQ, which is satisfiedif and only if the inhomogeneous minimum ofQ is bounded by a constant, sayµ(Q) ≤ 1. For this it iscrucial to observe thatxtQy is a linear expression in the parameters(qij). The PQFQ = (qij) defines theinner product of a Euclidean space(Rd, (·, ·)) by (x,y) = x

tQy.From the article [DDRS70],§3, of Delone, Dolbilin, Ryshkov and Stogrin we can extract the following

Proposition which will be central in our further discussion.

Proposition 7.1. Let L = conv{0,v1, . . . ,vd} ⊆ Rd be ad–dimensional simplex. Then the radius of the

circumsphere ofL is at most1 with respect to(·, ·) if and only if the following linear matrix inequality (inthe parametersqij) is satisfied:

BRL(Q) =

4 (v1,v1) (v2,v2) . . . (vd,vd)(v1,v1) (v1,v1) (v1,v2) . . . (v1,vd)(v2,v2) (v2,v1) (v2,v2) . . . (v2,vd)

......

.... . .

...(vd,vd) (vd,v1) (vd,v2) . . . (vd,vd)

� 0.

In [DDRS70] Delone et al. used this Proposition as the key ingredient for showing that the set of PQFswhich determine a circumsphere ofL with radius of at mostR is convex and bounded (see Proposition 9.1).Because of the importance of the Proposition and for the convenience of the reader we give a short proofhere.

Proof. We make use of Cayley–Menger determinants. The Cayley–Menger determinant ofd + 1 pointsv0, . . . ,vd with pairwise distancesdist(vi,vj) is

CM(v0, . . . ,vd) =

∣∣∣∣∣∣∣∣∣

0 1 . . . 11 dist(v0,v0)

2 . . . dist(v0,vd)2

......

.. ....

1 dist(vd,v0)2 . . . dist(vd,vd)

2

∣∣∣∣∣∣∣∣∣.

The squared circumradiusR2 of the simplexL equals (see e.g. Proposition 9.7.3.7 in [Ber87])

R2 = −1

2·det

(dist(vi,vj)

2)0≤i,j≤d

CM(v0, . . . ,vd).

2http://www.stanford.edu/˜boyd/MAXDET.html

16

Replacingdist(x,y)2 by (x,x)− 2(x,y) + (y,y), usingv0 = 0, and performing elementary transforma-tions of the determinants turns the above formula into

R2 = −1

4·

∣∣∣∣∣∣∣∣∣

0 (v1,v1) (v2,v2) . . . (vd,vd)(v1,v1) (v1,v1) (v1,v2) . . . (v1,vd)

......

.... . .

...(vd,vd) (vd,v1) (vd,v2) . . . (vd,vd)

∣∣∣∣∣∣∣∣∣det ((vi,vj))1≤i,j≤d

.

If R ≤ 1, then

4 · det ((vi,vj))1≤i,j≤d +

∣∣∣∣∣∣∣∣∣

0 (v1,v1) (v2,v2) . . . (vd,vd)(v1,v1) (v1,v1) (v1,v2) . . . (v1,vd)

......

.... . .

...(vd,vd) (vd,v1) (vd,v2) . . . (vd,vd)

∣∣∣∣∣∣∣∣∣≥ 0,

which is equivalent to ∣∣∣∣∣∣∣∣∣

4 (v1,v1) (v2,v2) . . . (vd,vd)(v1,v1) (v1,v1) (v1,v2) . . . (v1,vd)

......

..... .

...(vd,vd) (vd,v1) (vd,v2) . . . (vd,vd)

∣∣∣∣∣∣∣∣∣≥ 0.

After reordering the columns of the above matrix (1 ↔ d + 1, 2↔ d, . . . ), we see that all main minors arenon–negative. Hence, the matrixBRL(Q) is positive semi–definite.

Corollary 7.2. For anyd–dimensional simplexL ⊆ Rd with vertex at0 andQ ∈ Sd>0 we have

|BRL(Q)| ≥ 0 ⇐⇒ BRL(Q) � 0.

Example 7.3. Let us compute the matrix linear inequalityBRL(Q) � 0, Q = ( q11 q21q21 q22 ), for the two–

dimensional simplexL = conv{(00

),(10

),(11

)}. We have

BRL(Q) =

4 q11 q11 + 2q21 + q22q11 q11 q11 + q21

q11 + 2q21 + q22 q11 + q21 q11 + 2q21 + q22

=

4 0 00 0 00 0 0

+ q11

0 1 11 1 11 1 1

+ q21

0 0 20 0 12 1 2

+ q22

0 0 10 0 01 0 1

.

For a Delone polytopeL not being a simplex the circumradius is less or equal1 if and only if, it is less orequal1 for anyd–dimensional simplexL′ with vertices invertL. Therefore we setBRL(Q) = BRL′(Q)in this case. Since a block matrix is semi–definite if and onlyif the blocks are semi–definite, we have

17

Proposition 7.4. Let Q = (qij) ∈ Sd>0 be a PQF and letL1, . . . , Ln be a representative system of allnon–equivalentd–dimensional Delone polytopes inDel(Q). Then

µ(Q) ≤ 1 ⇐⇒

BRL1(Q) 0 0 . . . 0

0 BRL2(Q) 0 . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 . . . BRLn(Q)

� 0.

8 Algorithms

In this section we present algorithms which solve the lattice covering problem and the lattice packing–covering problem in any dimensiond.

Our algorithm for the lattice covering problem computes alllocally optimal lattice coverings of a givendimension. These are only finitely many because we will see that for every fixed Delone triangulationDthere exists at most one PQF lying in the topological closureof the secondary cone ofD giving a locallyoptimal covering density. So, we fix a Delone triangulation and try to find the PQF form which minimizesthe density function in the topological closure of the secondary cone of the fixed Delone triangulation. Wewill formulate this restricted lattice covering problem asa determinant maximization problem.

Our algorithm for the lattice packing–covering problem operates similarly. For every Delone triangula-tion and for every edge of the triangulation we have to solve asemidefinite programming problem.

8.1 Solving the Lattice Covering Problem

Recall that the covering density of a PQFQ in d variables is given byΘ(Q) =√

µ(Q)d

detQ · κd. Scaling ofQby a positive real numberα leavesΘ invariant. Consequently, we can restrict our attention to those PQFsQwith µ(Q) = 1. Hence, we solve the lattice covering problem if we solve theoptimization problem

maximize det(Q)subject to Q is a positive definite matrix,

µ(Q) = 1,

where the optimization variablesqij are the entries of the PQFQ. The major disadvantage of this optimiza-tion problem is that the second constraint is not expressible as a convex condition in the variablesqij andthat the problem has many local maxima. A locally optimal solution is also called alocally optimal latticecovering.

We will circumvent this by splitting the original problem into a finite number of determinant maximiza-tion problems. For every Delone triangulationD we solve the optimization problem

maximize det(Q)subject to Q is a positive definite matrix,

Q ∈∆(D),µ(Q) ≤ 1.

The relaxation of no longer requiringµ(Q) = 1 in the third constraint does not give more optimal solutionsbecause withQ also 1

µ(Q)Q satisfies the constraints. Now, we have to show that this is indeed a determinant

18

maximization problem. We have seen in Theorem 5.3 that the second constraint can be expressed withinequalities linear inqij. The constraintµ(Q) ≤ 1 can be expressed by a linear matrix inequality as inProposition 7.4.

The optimization vectorx ∈ Rd(d+1)/2 is the vector of coefficients ofQ. The linear matrix inequality

G(x) ≻ 0 is given byG(x) = Q. We encode the two other constraintsQ ∈∆(Q) andµ(Q) ≤ 1 by blockmatrices in the linear matrix inequalityF (x) � 0. For any linear inequality which is needed to describe thesecondary cone we have a1 × 1 block matrix. For any non–equivalentd–dimensional simplexL ∈ D wehave the(d+ 1)× (d+ 1) block matrixBRL(Q).

8.2 Solving the Lattice Packing–Covering Problem

Along the same lines as above we formulate the lattice packing–covering problem as a finite number ofsemidefinite programming problems.

Recall that the packing–covering constant of a PQFQ is γ(Q) = 2 ·√

µ(Q)λ(Q) . Sinceγ is homogeneous

we can assumeµ(Q) = 1 again, and the lattice packing–covering problem is equivalent to the followingoptimization problem.

maxmize λ(Q)subject to Q is a positive definite matrix,

µ(Q) = 1.

A locally optimal solution of the optimization problem is called a locally optimal lattice packing–covering.How do we computeλ(Q)? We say thatv ∈ Z

d\{0} is a shortest vectorof Q if Q[v] = λ(Q). ATheorem of Voronoi (cf. [Vor08], or [CS88], Chapter 21, Theorem 10) implies that a shortest vectorv givesthe edge[0,v] in the Delone subdivision ofQ. In a fixed Delone subdivisionD there are only finitely many(at most2(2d − 1)) edges of the form[0,v]. We can maximizeλ(Q) for Q ∈ ∆(D) as follows: For everyedge[0,v] ∈ D we maximizeQ[v] subject toQ[v] ≤ Q[w] wherew runs over all edges[0,w] ∈ D. In thisway we certify thatv is a shortest vector ofQ. All these expressions are linear in the coefficientsqij of Q.Hence we have to solve the following semidefinite programming problem for every non-equivalent DelonetriangulationD and for every edge[0,v] of D, in order to solve the lattice packing–covering problem:

maximize Q[v]subject to Q is a positive definite matrix,

Q ∈∆(D),µ(Q) ≤ 1,v is a shortest vector ofQ.

9 Properties of Local Optima

Since we are dealing with convex optimization problems, we can extract several structural information aboutuniqueness and invariance properties of locally optimal solutions. On the one hand these help us to identifyexact coordinates of optimal solutions. On the other hand they allow us to decide whether we have found anisolated locally optimal solution or not.

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9.1 Convexity and Invariance of Sets of Local Optima

Most of the desired properties follow from the special structure of sets with PQFsQ attaining certain fixedvaluesµ(Q), λ(Q) or det(Q). LetL = conv{0,v1, . . . ,vd} ⊆ R

d denote ad–dimensional simplex andDa Delone triangulation. Then we consider the sets

VL = {Q ∈ Sd>0 : |BRL(Q)| ≥ 0},VL = {Q ∈ Sd>0 : |BRL(Q)| = 0},WD =

⋂

L∈DVL ∩∆(D).

Hence, by Corollary 7.2 and Proposition 7.4 the PQFsQ ∈WD are those withµ(Q) ≤ 1 in ∆(D). Further,for D > 0 andλ > 0 we consider

DD = {Q ∈ Sd>0 : det(Q) ≥ D},Mλ = {Q ∈ Sd>0 : λ(Q) ≥ λ}.

The following is known (cf. [DDRS70], [Rys74]) and was partially discussed in previous sections:

Proposition 9.1. Let L = conv{0,v1, . . . ,vd} ⊆ Rd be ad–dimensional simplex,D a Delone triangula-

tion,D > 0 andλ > 0. Then

1. VL, and thereforeWD, is convex and bounded.

2. VL is a regular surface.

3. DD is unbounded and strictly convex.

4. Mλ is unbounded and convex and has a piecewise linear boundary.

5. DD andMλ are invariant with respect to the action ofGLd(Z) onSd>0.

As a consequence of these properties, it is not hard to derive(cf. [DDRS70], [Rys74])

Proposition 9.2. LetD be a Delone triangulation.

1. The set of PQFs∆(D) attainingminQ∈∆(D)

Θ(Q) is a single PQF, together with all of its positive

multiples. It is invariant with respect to the action ofD’s automorphism groupG ⊆ GLd(Z).

2. The set of PQFs∆(D) attainingminQ∈∆(D) γ(Q) is convex and contains a subset which is invariantwith respect to the action ofD’s automorphism groupG ⊆ GLd(Z).

A first analytic proof of property 1 was given by Barnes and Dickson [BD67]. There they also observed,as Ryshkov [Rys74] did for the lattice packing–covering problem, that it suffices to optimize among allPQFs whose automorphism group contains the groupG.

Corollary 9.3. If Q is a local optimum among all PQFs in∆(D) whose automorphism group containsG,thenQ is a locally optimal solution.

Note that the statement above holds only ifQ lies in theinterior of the secondary cone. For local optimaon the boundary of secondary cones we may apply the trivial

20

Proposition 9.4. A PQFQ is a locally optimal solution with respect toΘ or γ if and only if it is a locallyoptimal solution for all Delone triangulationsD with Q ∈∆(D).

By applying the foregoing Proposition we find e.g. two locally optimal solutions with respect toγ indimension 5 which both lie on the boundary of some secondary cones (see Section 11). At the momentthere is no evidence that there does exist a PQF on the boundary of some secondary cones, giving a locallyoptimal lattice covering. But we have a conjecture.

Conjecture 9.5. The PQF associated to the Leech lattice, which lies on an extreme ray of a secondary cone,is a locally optimal lattice covering.

9.2 Sufficient Conditions for Local Optima

Next we assume a PQFQ is given and we want to decide computationally if it is a locally optimal solution tothe lattice covering or lattice packing–covering problem.In both cases we can use thatQ with µ(Q) = 1 is alocally optimal solution if and only if there exists a seperating hyperplane ofWD andDdet(Q), respectivelyMλ(Q), throughQ. This can be checked by considering the outer normal cones ofthe convex sets at theircommon boundary pointQ. As a general reference to the basic concepts in convex and differential geometryused in the sequel we refer to the book [Sch93] by Schneider.

To be more precise, we considerSd as a Euclidean space with inner product〈·, ·〉. LetH = {S ∈ Sd :〈N,S〉 = α} be a hyperplane with normalN ∈ Sd \ {0} andC a convex set with boundary pointQ. ThenH is said to be a supporting hyperplane ofC atQ with outer normalN , if C ⊆ H− = {S ∈ Sd : 〈N,S〉 ≤〈N,Q〉}. The outer normal cone ofC atQ is then given by all outer normals of supporting hyperplanesatQ. Note that the outer normal cone is itself a convex cone. A hyperplaneH is called a seperating hyperplanefor two convex sets with a common boundary pointQ, if it is a supporting hyperplane of both sets, but withopposite outer normals. Such a hyperplane exists if and anlyif the corresponding outer normal conesN1

andN2 atQ satisfy−N1 ∩ N2 6= ∅.Because of the convexity ofDdet(Q), Mλ(Q) andWD a seperating hyperplane at a boundary pointQ of

WD and one of the other two sets yields a sufficient condition forQ to be a locally optimal solution to eitherthe lattice covering or the lattice packing–covering problem. Therefore a sufficient condition can be derivedfrom the outer normal cones atQ with respect to these sets. In both cases we need the outer normal cone ofWD atQ, which turns out to be polytopal, that is, it is the positive hull of only finitely many extreme rays:

Proposition 9.6. Let D be a Delone triangulation andQ ∈ WD a PQF withµ(Q) = 1. Then the outernormal cone ofWD atQ is equal to

− cone{gL(Q) : Q ∈ VL, L ∈ D},

wheregL(Q) = grad |BRL|(Q) denotes the gradient of the regular surfaceVL atQ.

Proof. First,−gL(Q) 6= 0 exists and is the unique outer normal ofVL atQ, sinceVL is a regular surfacedefined by the polynomial equation|BRL(Q)| = 0. Second, we see thatWD, in a sufficiently smallneighborhood ofQ, is equal to the intersection

⋂Q∈VL

VL. Then the outer normal cone ofWD at Q is

equal tocone{−gL(Q) : Q ∈ VL} (cf. [Sch93], 2.2) and the assertion follows.

It is easily checked that the outer normal cone ofDdet(Q) atQ is equal to the ray{−λQ−1 : λ > 0}.Thus the convexity ofDD andWD (cf. [DDRS70]) yields the following result by Barnes and Dickson[BD67]:

21

Proposition 9.7. Let D be a Delone triangulation. ThenQ ∈ ∆(D) is a locally optimal solution to thelattice covering problem if and only if

Q−1 ∈ − cone{gL(Q) : Q ∈ VL}.

A corresponding statement for the lattice packing–covering problem requires the knowledge of the outernormal cone ofMλ at a boundary pointQ. It is determined by the shortest vectors ofQ, that is, by thosev ∈ Z

d for which the homogeneous minimumλ(Q) is attained. Recall thatQ[v] = λ for a fixedv ∈ Zd\{0}

is linear in the parametersqij, hence this condition defines a hyperplane inSd. A normalVv ∈ Sd \ {0}of this hyperplane, henceVv with Q[v] = 〈Q,Vv〉, depends on the chosen inner product. Then, the outernormal cone ofMλ atQ is given by− cone {Vv : 〈Q,Vv〉 = λ(Q)}. (E.g., for practical computations it isconvenient to identifySd with R

d(d+1)/2 and to use the standard inner product〈Q,V 〉 = ∑i≤j qijvij and

Vv = (vij) with vij = (2− δij)vivj.) We conclude:

Proposition 9.8. LetD be a Delone triangulation. A PQFQ ∈ ∆(D) with µ(Q) = 1 is a locally optimalsolution to the lattice packing–covering problem if and only if

cone {Vv : 〈Q,Vv〉 = λ(Q)} ∩ − cone{gL(Q) : Q ∈ VL

}6= ∅.

Note again, that the two foregoing Propositions do not give criteria for PQFsQ on the boundary of somesecondary cones. In such a case we need to replace− cone{gL(Q) : Q ∈ VL} by a generalized expression.That is done by considering for each Delone triangulationD with Q ∈∆(D) the outer normal coneCD(Q)

of WD atQ. We haveCD(Q) = − cone(N1 ∪ N2) with normalsN1 = {gL(Q) : Q ∈ VL, L ∈ D}, asconsidered before, and normals

N2 ={N ∈ Sd : 〈N, ·〉 = (L,L′)(·), (L,L′) pair of adjacent simplices ofD, (L,L′)(Q) = 0

}

from facets of∆(D) containingQ.

Proposition 9.9. LetQ ∈ Sd>0. Then

1. Q is a locally optimal solution to the lattice covering problem if and only if

Q−1 ∈⋂

Q∈∆(D)

CD(Q).

2. Q is a locally optimal solution to the lattice packing–covering problem if and only if

cone {Vv : 〈Q,Vv〉 = λ(Q)} ∩⋂

Q∈∆(D)

CD(Q) 6= ∅.

Proof. The set⋂

Q∈∆(D) CD(Q) contains all common outer normals of the setsWD with Q ∈ ∆(D) atQ.

Thus⋂

Q∈∆(D)CD(Q) is empty, if and only if

⋃Q∈∆(D)

WD is not convex atQ. In this case there is notangent hyperplane of the latter set atQ and no local optimum for one of the convex optimization problems.If⋃

Q∈∆(D)WD is convex atQ though, then the existence of a seperating hyperplane with respect to the

convex setDdet(Q), respectivelyMλ(Q), gives a necessary and sufficient condition for a locally optimalsolution.

22

9.3 Conditions for Isolated Local Optima

Locally optimal solutions to the lattice packing–coveringproblem may not be isolated optima, in contrast tosolutions to the lattice covering problem. To determine computationally if a given locally optimal solutionQ is isolated, we can check if there exists a segment[Q,Q′] in the intersection ofMλ(Q) and WD, orrespectively in their boundaries. Thus there exists a separating hyperplane containingQ′ andQ, implyingin particular that the intersection in Proposition 9.8 doesnot contain interior points. Both sets are seperatedby a hyperplane with normalQ′ − Q, because possible “directions”S ∈ Sd with Q′ = Q + ǫS, ǫ > 0, inthe boundary ofMλ(Q) satisfy〈Vv, S〉 ≥ 0 for all v ∈ Z

d with Q[v] = λ(Q), as well as〈gL(Q), S〉 ≥ 0

for all L ∈ D with Q ∈ VL. Here, equality is attained at least once in both cases, since otherwise the setsMλ(Q) andWD would intersect in their interior.

SinceMλ(Q) has a piecewise linear boundary, equality〈Vv, S〉 = 0 implies thatQ′ = Q + ǫS is in theboundary ofMλ(Q) for sufficiently smallǫ > 0. Equality〈gL(Q), S〉 = 0 however implies only thatS is inthe tangent space ofVL atQ. ThusQ+ ǫS is in the boundary ofWD for sufficiently smallǫ > 0 if and onlyif the corresponding regular surfacesVL have curvature0 in directionS. This is equivalent toS being aneigenvector of eigenvalue0 of the HessianhL(Q) = hess |BRL|(Q). Hence, we have to check additionallyif the linear equationshL(Q)S = 0 are satisfied for allL ∈ D with 〈gL(Q), S〉 = 0.

On the other hand, if all these conditions are satisfied, a sufficiently smallǫ guarantees thatQ′ = Q+ǫSis contained in the common part of the boundaries ofMλ(Q) andWD.

Proposition 9.10. LetD be a Delone triangulation and letQ ∈ ∆(D) be a locally optimal solution to thelattice packing–covering problem. ThenQ is not an isolated local optimum if and only if there exists aS ∈ Sd with

1. 〈Vv, S〉 ≥ 0 for all v ∈ Zd with Q[v] = λ(Q)

2. 〈gL(Q), S〉 ≥ 0 for all L ∈ D with Q ∈ VL

3. hL(Q)S = 0 for all L ∈ D with Q ∈ VL and〈gL(Q), S〉 = 0

Note again, that the condition applies only ifQ is in the interior of a secondary cone. A similar conditionfor PQFs on the boundary of secondary cones can be attained byreplacinggL(Q) in 2 and 3 with the raysof the outer normal cone

⋂Q∈∆(D) CD(Q) of

⋃Q∈∆(D) WD atQ (cf. Proposition 9.9), if such exist.

Finally, we propose a (as far as we know) still unanswered

Problem 9.11. Does there exist a locally optimal solution to the lattice packing–covering problem whichis not isolated? And, does there exist a (locally optimal) solution to the lattice–packing covering problem,which does not have all the symmetries of the corresponding Delone subdivision?

10 Local Lower Bounds via Moments of Inertia

In this section we give simple and efficiently computable local lower bound of the lattice covering densityand the lattice packing–covering constant. These bounds are “local” in the sense that they only apply tothose PQFs lying in the topological closure of the secondarycone of a given Delone triangulation. For theircomputation we only need to know the coordinates of the simplices of the considered Delone triangulation.They are therefore useful tools in a heuristic search for “good” PQFs. The method goes back to Ryshkov andDelone. It is called the method of the moments of inertia because the central idea in its proof is analogousto the Parallel Axis Theorem in classical mechanics.

23

Let P ⊆ Rd be a finite set of points ind–dimensional Euclidean space(Rd, (·, ·)). We interpret the

points ofP as masses with unit weight. Themoment of inertiaof the points about a pointx ∈ Rd is defined

asIx(P ) =∑

v∈P dist(x,v)2. Thecentroidof P (center of gravity) is given bym = 1|P |

∑v∈P v. From

the equations

dist(x,v)2 = (x− v,x− v)= (x−m,x−m) + (m− v,m− v) + 2(x−m,m− v)= dist(x,m)2 + dist(m,v)2 + 2(x−m,m− v),

and∑

v∈P (m− v) = 0, we derive Apollonius’ formula (cf. [Ber87]§9.7.6)

Ix(P ) = |P |dist(x,m)2 + Im(P ). (1)

Hence, the moment of inertia about the centroidm is minimal.If the points ofP form the vertices of ad–dimensional simplex, then (1) gives a relationship between

the radius of the circumsphereR, the center of the circumspherec, and the moment of inertia about thecentroidm of P :

R2 =Ic(P )

d+ 1= dist(c,m)2 +

Im(P )

d+ 1.

We can computeIm(P ) using only the edge lengths of the simplexP . For everyw ∈ vertP we haveby definitionIw(P ) =

∑v∈P dist(w,v)2. Summing up and using (1) gives

∑

w∈PIw(P ) =

∑

w∈P((d + 1) dist(w,m) + Im(P )) = 2(d+ 1)Im(P ).

So, we get

Im(P ) =1

d+ 1

∑

{v,w}⊆P

dist(v,w)2. (2)

Let D be a Delone triangulation ofRd, let L1, . . . , Ln be thed–dimensional simplices of the star of alattice point (say e.g. the origin), and letmi be the centroid ofLi, i = 1, . . . , n. The arithmetical mean ofthe moments of inertia about the centroids ofLi with respect to a PQFQ is defined as

ID(Q) =1

n

n∑

i=1

Imi(Li),

and is called thecentral moment of inertiaof D with respect toQ. Note that we are now dealing with thescalar product given byQ with dist(x,y)2 = Q[x− y] and thatID(Q) is linear in the parametersqij of Q.

Proposition 10.1. The central moment of inertia ofD with respect toQ yields a lower bound of the inho-mogeneous minimum ofQ if D is a refinement ofDel(Q). In this case we have

µ(Q) ≥ 1

d+ 1ID(Q).

Proof. LetRi be the radius andci be the center of the circumsphere of the simplexLi, then

µ(Q) = maxi=1,...,n

R2i = max

i=1,...,n

(dist(ci,mi)

2 +Imi

(Li)

d+1

)

≥ maxi=1,...,n

Imi(Li)

d+1 ≥ 1(d+1)n

n∑i=1

Imi(Li)

= 1d+1ID(Q).

24

To find lower bounds for the lattice covering density or for the lattice packing–covering constant, wehave to minimize the linear functionID over all PQFs with a fixed determinant or with a fixed homogenousminimum respectively. As in previous sections we considerSd as a Euclidean space with inner product〈·, ·〉. Then we have

Proposition 10.2. There exists a PQFF ∈ Sd>0 with ID = 〈F, ·〉.Proof. SinceID is a linear function there is aF ∈ Sd with ID(·) = 〈F, ·〉. For every PQFQ we have〈F,Q〉 = ID(Q) > 0. SinceSd>0 is a self–dual cone, we haveF ∈ Sd>0.

Because of the preceding Proposition, minimizing the linear function ID, hence〈F, ·〉, over all PQFswith a fixed determinantD is geometrically equivalent to finding the unique PQF on the boundary ofDD

— the determinant–D–surface — that has a supporting hyperplane with normalF (see Section 9). Thus wederive (cf. [Val03] for a detailed proof)

Proposition 10.3. A linear functionf(·) = 〈F, ·〉 with F ∈ Sd>0 has a unique minimum on the determinant–D–surface. Its value isd d

√D detF and the minimum is attained at the PQFd

√D detFF−1.

Now we can plug Propositions 10.1 and Proposition 10.3 together. This yields a local lower bound forthe covering density of a PQF.

Proposition 10.4. LetD be a Delone triangulation. LetQ be a PQF for whichD is a refinement ofDel(Q).Then we have a lower bound for the covering density ofQ:

Θ(Q) ≥ Θ∗(D) =√(

d

d+ 1

)d

detF · κd,

whereF is the positive definite matrix given by the equationID(·) = 〈F, ·〉. Here and in the sequel wedenote the local lower bound for the Delone triangulationD by Θ∗(D).Proof. By Proposition 10.3,ID = 〈F, ·〉 has the unique minimumd d

√D detF on the determinant–D–

surface. Using this withD = detQ and applying Proposition 10.1 we get the lower boundΘ∗(D) for Θ(Q),because of

(Θ(Q)/κd)2 =

µ(Q)d

detQ≥

(ID(Q)

d+ 1

)d

/detQ ≥ dd detQ detF

(d+ 1)d detQ=

(d

d+ 1

)d

detF.

For a local lower bound on the lattice packing–covering constant we minimize the linear functionID(·) = 〈F, ·〉 over all PQFs with a fixed homogeneous minimum. In analogy to the above, we replacethe determinant–D–surface (the boundary ofDD) by the homogeneous–minimum–λ–surface (the bound-ary ofMλ).

Proposition 10.5. Let λ be a positive number. LetD be a Delone triangulation and letQ be a PQF forwhich D is a refinement ofDel(Q). Further, let the minimum of the linear functionID(·) = 〈F, ·〉 onthe homogeneous–minimum–λ–surface be attained at a PQFQF,λ Then we have a lower bound for thepacking–covering constant ofQ:

γ(Q) ≥ 2

√〈F,QF,λ〉(d+ 1)λ

.

25

Proof. The PQFλ(Q)λ QF,λ attains the minimum of〈F, ·〉 among PQFs inMλ(Q). Applying Proposition 10.1

gives the desired lower bound:

(γ(Q)/2)2 =µ(Q)

λ(Q)≥

(ID(Q)

d+ 1

)/λ(Q) ≥ 〈F,

λ(Q)λ QF,λ〉

(d+ 1) · λ(Q)=〈F,QF,λ〉(d+ 1)λ

.

But practically it is difficult to obtainQF,λ and therefore the lower bound of the Proposition. Instead, ifwe minimizeID on an approximationMD

λ of Mλ, suitable forD, things become easier. We set

MDλ =

{Q ∈ Sd>0 : Q[v] ≥ λ for all v ∈ Z

d with [0,v] ∈ D}.

Now finding the minimum ofID on MDλ reduces to a linear program with finitely many constraints. By

looking at equation (2) it is easy to see that it is bounded below since every summand is at leastλ. Becauseof Mλ ⊆ MD

λ we get

Proposition 10.6. Let λ be a positive number. LetD be a Delone triangulation and letQ be a PQF forwhichD is a refinement ofDel(Q). Further, let the minimum of the linear functionID(·) = 〈F, ·〉 on MD

λ

be attained at a PQFQDF,λ. Then we have a lower bound for the packing–covering constant of Q:

γ(Q) ≥ γ∗(D) = 2

√〈F,QD

F,λ〉(d+ 1) · λ.

11 Computational Results

In Section 8 we developed algorithms for the solution of the lattice covering and the lattice packing–coveringproblem. Here, we want to demonstrate that these algorithmsare not purely of theoretical interest. Weimplemented the algorithms in C++, using the packageMAXDET3 of Wu, Vandenberghe and Boyd and thepackagelrs4 of Avis as subroutines. The computations were perfomed by a usual Intel Pentium basedcomputer. The implemented algorithm is able to solve both problems in dimensionsd = 1, . . . , 5 andhereby verifies the known results. With the help of a heuristic method it produces interesting new latticesin dimensionsd = 6 which give better coverings and packing–coverings than previously known ones.Unfortunately, in higher dimensions the implementation does not perform very well, mainly due to memorylimitations.

11.1 Dimensions 1, . . . , 5

We described the solutions to both problems for the1, 2, and3–dimensional cases already in Sections 3and 4. Since in these cases there is only one type of Delone triangulation, these results are computationallyrather trivial.

The 4–dimensional case requires more work because there are three non–equivalent Delone triangu-lations in dimension4. Baranovskii [Bar65] and independently Dickson [Dic67] found all three non–equivalent locally optimal solutions to the lattice covering problem in dimension4. Earlier Delone and

3http://www.stanford.edu/˜boyd/MAXDET.html4http://cgm.cs.mcgill.ca/˜avis/C/lrs.html

26

Ryshkov showed that the latticeA∗4 gives the best4–dimensional lattice coveringwithout determining all

locally optimal solutions. We will describe their approachin Section 11.2.For the lattice packing–covering problem the4–dimensional case was resolved by Horvath [Hor82]. He

shows, that there exist, as in the covering case, three isolated, locally optimal solutions — one for eachDelone triangulation. It is interesting that Horvath’s optimal packing–covering latticeHo4 does not belongto the family of root lattices and their duals. An associatedPQF is

QHo4 =

2 1 −1 −11 2 −1 −1−1 −1 2 0−1 −1 0 2

+

1

3

√3

4 1 −2 −21 4 −2 −2−2 −2 4 1−2 −2 1 4

.

The first summand, associated to the best packing latticeD4, lies on an extreme ray of the secondary conebelonging toQHo4 .

In a series of papers Ryshkov and Baranovskii solved the5–dimensional lattice covering problem. In[Rys73] Ryshkov introduced the concept of C–types. Two Delone triangulations are of the sameC–typeif their 1–skeletons (the graph consisting of vertices and edges of the triangulation) coincide. He gaves analgorithm to find all non–equivalent C–types in any given dimension. He computed that there are3 non–equivalent C–types in dimension4 and that there are76 non–equivalent C–types in dimension5. Using thislist Baranovskii and Ryshkov enumerated221 (of 222) non–equivalent5–dimensional Delone triangulationsin [BR73]. They described the triangulations in more detailin [BR75]. In the last paper of the series [RB75]they showed that the latticeA∗

5 provides the least dense5–dimensional lattice covering. In their proof they donot find all locally optimal lattice coverings. By using estimations (see Section 11.2) they merely show thatall local minima exceed the covering density ofA∗

5. As mentioned in Section 5.3, Ryshkov and Baranovskiimissed one Delone triangulation. Fortunately, this does not give a thinner lattice covering thanA∗

5.Using our algorithm, we produced a complete table of all non–equivalent locally optimal lattice cover-

ings in dimension5. The computation took about20 minutes on a standard Intel Pentium computer.

Theorem 11.1. In dimension5, there exist222 non–equivalent, local minima of the lattice covering densityfunctionΘ, ranging from≈ 2.124286 to≈ 2.821159. Thus all locally optimal solutions are attained in theinterior of their secondary cones.

Our list also shows that the PQF of Barnes and Trenerry [BT72]yields the second best locally optimallattice covering with density≈ 2.230115.

In 1986, Horvath [Hor86] solved the lattice packing–covering in dimension5. The proof is only availablein Russian and about 70 pages long (private communication).As in dimension4, Horvath’s latticeHo5 isnot among previously known ones. An associated PQF is

QHo5 =

2 1 0 −1 −11 2 −1 −1 00 −1 2 0 0−1 −1 0 2 −1−1 0 0 −1 2

+29− 7

√13

18√13− 60

6 3 −2 −2 −23 6 −2 −3 −2−2 −2 6 −1 −1−2 −3 −1 6 0−2 −2 −1 0 6

.

Again, the first summand is associated to the best packing latticeD4, lies on an extreme ray of the secondarycone belonging toQHo5 .

By applying Propositions of Section 9 we were able verify Horvath’s result and moreover to attain acomplete list of all locally optimal solutions. In contrastto the covering problem, there are many secondarycones that contain no locally optimal solution.

27

Theorem 11.2. In dimension5, there exist47 non–equivalent, local minima of the lattice packing–coveringconstantγ, ranging from≈ 1.449456 to≈ 1.645666. 45 of them are in the interior of their secondary conesand2 are each on the boundaries of3 non–equivalent ones.

Note that in contrast to the covering problem, we have not only secondary cones attaining a locallyoptimal solution on the boundary, but also many — namely171 — which do not contain a locally optimalsolution at all.

11.2 Heuristic Methods

Before going to dimension6, let us explain one important heuristic method, which is essential for findingthe new lattices. One of the biggest problem in finding good lattice coverings or packing–coverings indimension6 and higher is that it is not apriori clear which Delone triangulation admit good ones. Solvinga determinant maximization problem is a rather time consuming task and there are a lot of non–equivalentDelone triangulations to consider. So a desirable tool are fast computable lower bounds, as described inSection 10.

We view the set of Delone triangulations as an undirected labeled graph. A node represents a Delonetriangulation and two nodes are adjacent if their Delone triangulations are bistellar neighbours. LetD be aDelone triangulation. We label its node by the local lower boundΘ∗(D) or γ∗(D). We can use the labelingin two different ways.

On the one hand it is clear that if the labeling of a node is large, then the considered Delone triangulationdoes not admit a good lattice covering and lattice packing–covering respectively. Ryshkov and Delone[DR63] solved the lattice covering problem in dimension4 by this method. But in this way the five–dimensional lattice covering problem cannot be solved. From the222 non–equivalent Delone triangulationsthere are20 whose local lower bound is smaller thanΘ(A∗

5).On the other hand we can hope thatD admits a good lattice covering or packing–covering if the local

lower bound is small. In dimension6 the hope, that good local lower bounds yield good lattice coveringsis partially fulfilled. We report on a typical example: We started from the Delone triangulation of Voronoi’sprincipal form of the first type. We took a random walk of length 50. Then, we found a node labeledby Θ∗(D) ≈ 2.58514. From this we proceeded by taking a neighbouring node havingthe smallest locallower bound. By repeating this greedy strategy we resulted in a node labeled byΘ∗(D) ≈ 2.31802. Atthe moment this node is interesting for several “extremeness” properties. It yields the smallest known locallower bound and it has the largest known number of neighbours, namely130. As we will see in the nextsection there exists a locally optimal lattice covering which belongs to this node with covering density≈ 2.46612. At present this is the second best known6–dimensional lattice covering. Furthermore we willsee that there exists a locally optimal lattice packing–covering belonging to this node which currently definesthe best known6–dimensional lattice packing–covering.

11.3 New Six–Dimensional Lattices

Until now we were not able to solve the lattice covering problem in this dimension. As we reported inSection 3, Ryshkov [Rys67] asked what is the first dimensiond whereA∗

d does not give the least denselattice covering.

A classification of all non–equivalent Delone triangulations in dimension6 is not in reach. So we cannot use our algorithm to find the best lattice covering. Nevertheless we run it partially to find good latticecoverings. Since there are no other good lattice coverings in the neighbourhood of Voronoi’s principal form

28

of the first type, we used the heuristic method described in the previous section. In this manner we foundabout100 new lattice coverings which were better thanA∗

6. Thus, we can give an answer to Ryshkov’squestion:

Theorem 11.3. Dimensiond = 6 is the smallest dimension where the latticeA∗d does not give the least

dense lattice covering.

Two of the new lattice coverings were strikingly good. Of course, by the computational optimization weonly got numerical estimates of these coverings.

The now second best–known lattice covering was quite easy tofind: running our heuristic methoddescribed above finds this lattice covering in most of the trials. An approximation is given by the PQF

Qc26 ≈

1.9982 0.5270 −0.4170 −0.5270 0.5270 −1.05410.5270 1.9982 −0.4170 −0.5270 0.5270 −1.0541−0.4170 −0.4170 2.1082 −1.0541 −0.4170 0.8341−0.5270 −0.5270 −1.0541 1.9982 −0.5270 −0.41700.5270 0.5270 −0.4170 −0.5270 1.9982 −1.0541−1.0541 −1.0541 0.8341 −0.4170 −1.0541 2.1082

and its covering density isΘ(Qc26 ) ≈ 2.466125. The Delone subdivision is a triangulation and its secondary

cone has130 facets. The local lower bound is≈ 2.318034.For a while we thought that this might by the best lattice covering in dimension6. But then

Qc16 ≈

2.0550 −0.9424 1.1126 0.2747 −0.9424 −0.6153−0.9424 1.9227 −0.5773 −0.7681 0.3651 −0.36511.1126 −0.5773 2.0930 −0.4934 −0.5773 −0.98040.2747 −0.7681 −0.4934 1.7550 −0.7681 0.7681−0.9424 0.3651 −0.5773 −0.7681 1.9227 −0.3651−0.6153 −0.3651 −0.9804 0.7681 −0.3651 1.9227

with covering densityΘ(Qc16 ) ≈ 2.464802 came up. The Delone subdivision is a triangulation and its

secondary cone has100 facets. The local lower bound is≈ 2.322204. After this, we did not found anyfurther lattice covering records in dimension6. Furthermore, we did not find a6–dimensional Delonetriangulation whose secondary cone has more than130 facets or whose local lower bound is less than2.318034.

Nevertheless, in the secondary cone ofQc26 we found the PQF

Qpc6 ≈

2.0088 0.5154 0.5154 −0.5154 0.9778 0.51540.5154 2.0088 0.5154 −0.5154 −0.5154 −0.97780.5154 0.5154 2.0088 −0.5154 −0.5154 0.5154−0.5154 −0.5154 −0.5154 2.0088 −0.9778 −0.51540.9778 −0.5154 −0.5154 −0.9778 2.0088 0.97780.5154 −0.9778 0.5154 −0.5154 0.9778 2.0088

,

which gives currently the best known lattice packing–covering in dimension6 with packing–covering con-stantγ(Qpc

6 ) ≈ 1.411081. In the next subsection we will examine these new PQFs in greater detail.

29

11.4 Beautification and a Unified View

Although we found an answer to Ryshkov’s question, these results are not fully satisfying. We want to knowthe exact lattices. Even more important, we want to know an interpretation of why these lattice covering aregood. To accomplish this we collect some more data.

The automorphism group ofDel(Qc26 ) has order3840 and the one ofDel(Qc1

6 ) has order240. Withthe knowledge of the groups we were able to compute the extreme rays of both secondary cones [DV04].The secondary coneC1 = ∆(Del(Qc2

6 )) has7, 145, 429 and the coneC2 = ∆(Del(Qc16 )) has2, 257, 616

extreme rays. Both contain an extreme ray associated to the latticeE∗6 given e.g. by the PQF

QE∗

6=

4 1 2 2 −1 11 4 2 2 2 12 2 4 1 1 22 2 1 4 1 2−1 2 1 1 4 21 1 2 2 2 4

.

After transformingQc16 andQc2

6 by integral unimodular transformations we can assume that the Delone tri-angulations of the two PQFs are refinements of the Delone subdivision Del(QE∗

6). This Delone subdivision

was investigated in different contexts (cf. [Wor87], [CS91], [MP95] and [Bar92]). We briefly describe themain results: The covering density isΘ(QE∗

6) = 8

9√3·κ6 ≈ 2.652071. All 6–dimensional Delone polytopes

of QE∗

6are pairwise congruent. In the star of the origin there are720 full–dimensional Delone polytopes,

and the automorphism group ofQE∗

6acts transitively on these full–dimensional polytopes. Each polytope

is the convex hull of three triangles lying in three pairwiseorthogonal affine planes. Each polytope has9vertices,27 facets, and three different triangulations, where each triangulation consists of9 six–dimensionalsimplices.

Using this information we are able to prove

Theorem 11.4.The PQFQc16 gives the least dense lattice covering among all PQFs whose Delone subdivi-

sion is a refinement of the Delone subdivisionDel(QE∗

6).

Proof. Our proof is again computational and uses a branch and cut method. We have to show that thesecondary cones of all Delone triangulations refiningDel(QE∗

6) do not contain a PQF with covering density

less thanΘ(Qc16 ). There are40 full–dimensional Delone polytopes ofQE∗

6which cannot be transformed into

each other by translations or by the mapx 7→ −x. Since every of these Delone polytopes has3 possibletriangulations, the number of all periodic triangulationsrefining Del(QE

∗

6) is 340. It is not apriori clear

how to distinguish between Delone and non–Delone triangulations, and it is not possible to generate all340

triangulations. We choose a backtracking approach instead.We arrange partial triangulations in a tree. On every level one of the40 Delone polytopes is triangulated

so that we have3n nodes on then–th level. For every nodeN we define the value

ΘN = max{det(Q) : Q ∈ S6>0, BRL(Q) � 0 for all simplicesL of partial triangulationN

}.

Obviously,ΘN is a lower bound for the covering density of any PQF whose Delone subdivision refines thepartial triangulationN . We can compute this value by solving a determinant maximization problem similarto the one in Section 8.1. Note thatBRL(Q) � 0 is equivalent toQ ∈ VL (cf. Section 9). By a depth–firstsearch in the tree we compute the smallest lower bound for every level. At level23 the smallest lower boundequals the covering density ofQc1

6 , so we can stop the procedure there.

30

The knowledge of the automorphism groups ofQc16 andQc2

6 also enables us to give a unified view onboth lattices. For their automorphism groups we have

Aut(Qc16 ) ⊆ Aut(Qc2

6 ) ⊆ Aut(QE∗

6).

Aut(Qc26 ) turns out to be the subgroup ofAut(QE∗

6) stabilizing the minimal vectors±e1, andAut(Qc1

6 )is the intersection of the two subgroups ofAut(QE∗

6) stabilizing the minimal vectors±e1 and±e2 respec-

tively.The subspaceI1 of all quadratic forms invariant under the groupAut(Qc2

6 ) is spanned by the PQFsQE∗

6

andR1 (see below). At the same time,I1∩C1 is a cone with extreme raysQE∗

6, R1. By Proposition 9.2,Qc2

6

has to lie incone{QE∗

6, R1}. The subspaceI2 of all quadratic forms invariant under the groupAut(Qc1

6 ) isfour–dimensional. The coneI2 ∩C2 has six extreme raysQE∗

6, R2, . . . , R6.

R1 =

12 3 6 6 −3 33 7 4 4 3 26 4 8 3 1 46 4 3 8 1 4−3 3 1 1 7 33 2 4 4 3 7

R2 =

0 0 0 0 0 00 5 2 2 3 10 2 4 0 2 20 2 0 4 2 20 3 2 2 5 30 1 2 2 3 5

R3 =

6 4 4 4 0 24 11 6 6 5 34 6 8 3 3 44 6 3 8 3 40 5 3 3 7 42 3 4 4 4 7

R4 =

3 2 2 2 0 12 3 2 2 1 12 2 4 1 1 22 2 1 4 1 20 1 1 1 3 21 1 2 2 2 4

R5 =

7 3 4 4 −1 23 17 8 8 9 44 8 12 3 5 64 8 3 12 5 6−1 9 5 5 13 72 4 6 6 7 12

R6 =

9 6 6 6 0 36 9 6 6 3 36 6 8 4 2 46 6 4 8 2 40 3 2 2 5 33 3 4 4 3 6

Note thatR2 lies in I1. Altogether, this yields the a “picture” in dimension21 given in Figure 5.Let us finally try to find the exact coordinates of the the PQFs.This is easy forQc2

6 . We know that we canscaleQc2

6 so thatQc26 = QE∗

6+ xR1, for somex ∈ R≥0. Now, the exact finding ofx boils down to finding

roots of an univariate polynomial: we have to minimize the functionx 7→ µ(QE∗

6+xR1)

d/det(QE∗

6+xR1)

where we know, because of the approximate solution, thatµ(QE∗

6+ xR1) is a polynomial for all points in a

sufficiently small neighbourhood of the exactx. This calculus exercise leads to

Qc26 = QE∗

6+

√1057 − 1

88R1,

and

Θ(Qc26 ) = 1323

√2(1543 + 41

√1057)3

(5√1057 + 259)3

√23836111129

√1057 + 774881898455

· κ6

≈ 2.466121650.

What is the general pattern behind this beautification process? LetQ be a locally optimal lattice coveringwith Delone triangulationD. We use the symmetry ofD to find the subspace in whichQ lies. This reducesthe number of involved variables. The simplices of the Delone triangulation which have circumradius1 giveequality constraints. Then we maximize the determinant of the quadratic forms lying in the subspace subject

31

QE∗

6

R1

R2

Qc16

Qc26

Qpc6

C1

C2

invariant subspaceI1

boundary ofS6≥0

Figure 5. Unified view onQE∗

6, Qc1

6 ,Qc26 andQpc

6 .

to the equality constraints. For this optimization problem, which involves only algebraic equations, we canuse Grobner basis techniques.

Unfortunately, we were not able to solve the corresponding algebraic equations forQc16 . By the method

explained above we computeQc16 = QE∗

6+ xR2 + yR3 + zR4 where

−35yz3 − 6y2z + 4y2z2 + 5y3z − 15z2 + 6y3− 11yz − 25yz2 + 8y2

− 10z4 − 21z3 − 4z + 2y

14y3− 31y2z − 6z3 + 6y − 11z + 20y2

− 34yz − 15z2 − 31yz2= x

3y4 + (−7z + 7)y + (−14z − 9z2 + 5)y2 + (−9− 9z3 − 14z2 + 1)y − 5z2 − 2z − 5z3 − 2z4 = 0

andz is the root near0.138317657 of ∂Θ(z)∂z

where

Θ(z) =1

4096(72 + 1862xy2z + 2508xy2 + 264x + 520y + 861x2yz + 809xyz2 + 2208xyz + 941yz2

+1436xy3 + 788y2z2 + 1238yz + 223yz3 + 2143y2z + 1217y3z + 117xz3 + 487xz2 + 216x3y

+108x3z + 966x2y2 + 521x2z + 1166x2y + 193x2z2 + 634xz + 1424xy + 232z + 1380y2

+1622y3 + 270z2 + 22z4 + 132z3 + 332x2 + 710y4 + 144x3)6/

((6xy + 3xz + 4x+ 12y + z2 + 4z + 7yz + 10y2 + 3)7(4x+ 5y + 3z + 3)10)

At least this gives a possibility to approximateQc16 .

Using a similar and more successful beautification process for Qpc6 , we find

Qpc6 = QE∗

6+

√798 − 18

79R1,

with lattice packing–covering constant

γ(Qpc6 ) = 4

√7√

1083 + 32√7√114√

190 + 7√7√114

√5√7√114 + 147

≈ 1.411081242.

Using the tools described in Section 9 we get

32

Theorem 11.5.Qpc6 is an isolated, locally optimal solution to the lattice packing–covering problem, lying

in the interior of its secondary cone.

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Achill Schurmann, Department of Mathematics, University of Magdeburg, 39106 Magdeburg, Germany,email: achill@mathematik.uni-magdeburg.de

Frank Vallentin, Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904,Israel, email:vallenti@ma.tum.de

36