[fields institute monographs] lectures on sphere arrangements – the discrete geometric side volume...

Chapter 8Proofs on Coverings by Cylinders

Abstract First, we give a proof of the long-standing affine plank conjecture of Bangfor successive hyperplane cuts and then for inductive partitions. Second, we provea lower estimate for the sum of the cross-sectional volumes of cylinders covering aconvex body in Euclidean d -space. Then we prove a Kadets–Ohmann-type theoremin spherical d -space for coverings of balls by convex bodies via volume maximizinglunes. Finally, we give estimates for partial coverings of balls by planks in Euclideand -space.

8.1 Proof of Theorem 7.1.8

Theorem 7.1.8. Let K and C be convex bodies in Ed , d � 2 and let the positive

integer m be given. If K is sliced into n � 1 pieces by n � 1 successive hyperplanecuts (i.e., when each cut divides one piece), then the minimum of the greatest mthsuccessive C-inradius of the pieces is the .mn/th successive C-inradius of K (i.e.,rC.K; mn/). An optimal partition is achieved by n � 1 parallel hyperplane cutsequally spaced along the minimal C-width of the rC.K; mn/C-rounded body of K.

8.1.1 On Coverings of Convex Bodies by Two Planks

On the one hand, the following statement is an extension to higher dimensions ofTheorem 4 in [3]. On the other hand, the proof presented below is based on Theorem4 of [3].

Lemma 8.1.1. If a convex body K in Ed ; d � 2 is covered by the planks P1 and P2,

then wC.P1/ C wC.P2/ � wC.K/ for any convex body C in Ed .

Proof. Let H1 (resp., H2) be one of the two hyperplanes which bound the plankP1 (resp., P2). If H1 and H2 are translates of each other, then the claim is

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side,Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 8,© Springer International Publishing Switzerland 2013

143

144 8 Proofs on Coverings by Cylinders

obviously true. Thus, without loss of generality we may assume that L WD H1 \ H2

is a .d � 2/-dimensional affine subspace of Ed . Let E2 be the 2-dimensional linearsubspace of Ed that is orthogonal to L. If .�/0 denotes the (orthogonal) projection ofE

d parallel to L onto E2, then obviously, wC0.P0

1/ D wC.P1/, wC0.P02/ D wC.P2/

and wC0.K0/ � wC.K/. Thus, it is sufficient to prove that

wC0.P01/ C wC0.P0

2/ � wC0.K0/:

In other words, it is sufficient to prove Lemma 8.1.1 for d D 2. Hence, in the restof the proof, K; C; P1; P2; H1, and H2 mean the sets introduced and defined above,however, for d D 2. Now, we can make the following easy observation

wC.P1/ C wC.P2/ D w.P1/

w.C; H1/C w.P2/

w.C; H2/

D w.P1/

w.K; H1/

w.K; H1/

w.C; H1/C w.P2/

w.K; H2/

w.K; H2/

w.C; H2/

��

w.P1/

w.K; H1/C w.P2/

w.K; H2/

�wC.K/

D .wK.P1/ C wK.P2// wC.K/:

Then recall that Theorem 4 in [3] states that if a convex set in the plane iscovered by planks, then the sum of their relative widths is at least 1. Thus, usingour terminology, we have that wK.P1/ C wK.P2/ � 1, finishing the proof ofLemma 8.1.1. �

8.1.2 Minimizing the Greatest mth Successive C-Inradius

Let K and C be convex bodies in Ed , d � 2. We prove Theorem 7.1.8 by induction

on n. It is trivial to check the claim for n D 1. So, let n � 2 be given and assumethat Theorem 7.1.8 holds for at most n � 2 successive hyperplane cuts and based onthat we show that it holds for n � 1 successive hyperplane cuts as well. The detailsare as follows.

Let H1; : : : ; Hn�1 denote the hyperplanes of the n � 1 successive hyperplanecuts that slice K into n pieces such that the greatest mth successive C-inradius ofthe pieces is the smallest possible say, �. Then take the first cut H1 that slices K intothe pieces K1 and K2 such that K1 (resp., K2) is sliced into n1 (resp., n2) pieces bythe successive hyperplane cuts H2; : : : ; Hn�1, where n D n1 C n2. The inductionhypothesis implies that � � rC.K1; mn1/ DW �1 and � � rC.K2; mn2/ DW �2 andtherefore

wC.K1�C/ � wC.K1

�1C/ D mn1�1 � mn1�I (8.1)

moreover,

8.2 Proof of Theorem 7.1.12 145

wC.K2�C/ � wC.K2

�2C/ D mn2�2 � mn2�: (8.2)

Now, we need to define the following set.

Definition 8.1.2. Assume that the origin o of Ed belongs to the interior of the

convex body C � Ed . Consider all translates of �C which are contained in the

convex body K � Ed . The set of points in the translates of �C that correspond to o

form a convex set called the inner �C-parallel body of K denoted by K��C.

Clearly,

.K1/��C [ .K2/��C � K��C with .K1/��C \ .K2/��C D ;:

Also, it is easy to see that there is a plank P with wC.P/ D � such that it is parallelto H1 and contains H1 in its interior; moreover,

K��C � .K1/��C [ .K2/��C [ P:

Hence, applying Lemma 8.1.1 to .K1/��C as well as .K2/��C we get that

wC�K��C

� � wC�.K1/��C

�C � C wC�.K2/��C

�: (8.3)

It follows from the definitions that wC�.K1/��C

� D wC.K1�C/ � �,

wC�.K2/��C

� D wC.K2�C/ � � and wC

�K��C

� D wC.K�C/ � �. Hence, (8.3)is equivalent to

wC.K�C/ � wC.K1�C/ C wC.K2

�C/: (8.4)

Finally, (8.1),(8.2), and (8.4) yield that

wC.K�C/ � mn1� C mn2� D mn�: (8.5)

Thus, (8.5) clearly implies that rC.K; mn/ � �. As the case, when the optimalpartition is achieved, follows directly from the definition of the mnth successiveC-inradius of K, the proof of Theorem 7.1.8 is complete.


Theorem 7.1.12. Let K and C be convex bodies in Ed ; d � 2 and let m be a

positive integer. If V1 [ V2 [ � � � [ Vn is an inductive partition of Ed such thatint.Vi \ K/ ¤ ; for all 1 � i � n, then

nXiD1

rC.Vi \ K; m/ � rC.K; m/:


8.2.1 On an Extension of a Helly-Type Result of Klee

Recall the following Helly-type result of Klee [137]. Let F WD fAi j i 2 I g be afamily of compact convex sets in E

d ; d � 2 containing at least d C 1 members.Suppose C is a compact convex set in E

d such that the following holds: For eachsubfamily of d C 1 sets in F , there exists a translate of C that is contained in alld C1 of them. Then there exists a translate of C that is contained in all the membersof F . In what follows we give a proof of the following extension of Klee’s theoremto linear packings.

Theorem 8.2.1. Let F WD fAi j i 2 I g be a family of convex bodies in Ed ; d � 2

containing at least d C 1 members. Suppose C is a convex body in Ed and m � 1 is

a positive integer moreover, l is a line passing through the origin o in Ed such that

the following holds: For each subfamily of d C 1 convex bodies in F , there existsa linear packing of m translates of C with separating direction l that is containedin all d C 1 of them. Then there exists a linear packing of m translates of C withseparating direction l that is contained in all the members of F .

Proof. Let C1; : : : ; Cm be a linear packing of m translates of C with separatingdirection l in E

d . In what follows we always assume that whenever we writeC1; : : : ; Cm, then the orthogonal projections Prl .C1/; : : : ; Prl .Cm/ of the translatesC1; : : : ; Cm of C onto the line l , listed in this order, form a consecutive sequenceof congruent non-overlapping intervals with respect to some fixed orientation of l .Let Li denote the family of all linear packings of m translates of C with separatingdirection l which lie in Ai . Moreover, without loss of generality we may assumethat the origin o of Ed is in C. Now, for each Ai , i 2 I let

.Ai /�C WD ft 2 Ed j t C C is the first member of a linear packing in Li g:

Clearly, each .Ai /�C, i 2 I is a compact set. Furthermore, we claim that .Ai /�C isa convex set for all i 2 I . Indeed, it is rather straightforward to show that if

t1 C C; C2; : : : ; Cm�1; Cm; and t01 C C; C0

2; : : : ; C0m�1; C0

m

are linear packings chosen from Li , then there exists a linear packing

�t1 C .1 � �/t01 C C; C00

2 ; : : : ; C00m�1; �Cm C .1 � �/C0

m

of m translates of C with separating direction l contained in the convex hull

conv�.�t1 C .1 � �/t0

1 C C/ [ .�Cm C .1 � �/C0m/�

such that it is also in Li for any given 0 � � � 1. Thus, based on the assumptions ofTheorem 8.2.1, the family f.Ai /�C j i 2 I g is a collection of compact convex setsof Ed with the property that


.Ai1 /�C \ .Ai2 /�C \ � � � \ .AidC1/�C ¤ ;

for all pairwise distinct i1; i2; : : : ; idC1 2 I . Hence, Helly’s theorem (see forexample [17], p. 19) implies that \f.Ai /�C j i 2 I g ¤ ;, finishing the proof ofTheorem 8.2.1. �

8.2.2 On Some Concave Functions of Successive Inradii

A rather straightforward extension of the method of Akopyan and Karasev [2]combined with Theorem 8.2.1 gives the following statement. For the statementbelow as well as its proof we extend the definition of the mth successive C-inradiusof convex bodies K � E

d via including all non-empty compact convex sets K � Ed

having intK D ; with the definition rC.K; m/ WD 0 and via including the empty set; with the definition rC.;; m/ WD �1.

Theorem 8.2.2. Let K and C be convex bodies in Ed ; d � 2 and let m be a positive

integer. Moreover, let V1 [ V2 [ � � � [ Vn be an inductive partition of Ed and letKi .x/ WD K \ .x C Vi / for all x 2 E

d and 1 � i � n. Then the function

r.x; m/ WDnX

iD1

rC .Ki .x/; m/

is a concave function of x 2 Ed .

Proof. First, we need the following statement.

Lemma 8.2.3. Let P WD fx 2 Ed j Li .x/ � 0 for al l 1 � i � pg be an

arbitrary d -dimensional convex polytope of Ed defined by the linear inequalitiesfLi .x/ � 0; 1 � i � pg. If P.y/ WD fx 2 E

d j Li .x/ C yi � 0 for al l 1 � i � pgstands for the parallel deformation of P generated by the vector y D .y1; : : : ; yp/ 2E

p , then

rC .P.y/; m/

is a concave function of y for any fixed convex body C of Ed and for any fixedpositive integer m.

Proof. Let l be an arbitrary line passing through the origin o in Ed . Moreover, let

rC .P.y/; m j l/ denote the largest real � > 0 with the property that there existsa linear packing of m translates of �C lying in P.y/ and having l as a separatingdirection. Furthermore, let

PI .y/ WD fx 2 Ed j Li .x/ C yi � 0 for all i 2 I g


whenever I � Œp� WD f1; 2; : : : ; pg and let rC .PI .y/; m j l/ be defined similarly torC .P.y/; m j l/. The definition of successive inradii based on linear packings andTheorem 8.2.1 imply in a straightforward way that

rC .P.y/; m/ D info2l�Ed

rC .P.y/; m j l/ (8.6)

rC .P.y/; m j l/ D infI�Œp�; card.I /�dC1

rC .PI .y/; m j l/ (8.7)

Next, we take a closer look of the convex polyhedral set PI .y/: Clearly, PI .y/

is either a (convex polyhedral) cylinder (of some convex polyhedral base set havingdimension strictly less than d ), or a (convex polyhedral) cone, or a d -dimensionalsimplex. In the first case, we use induction on the dimension of the base set. In thesecond case, we always have that rC .PI .y/; m j l/ D C1. In the third case, it iseasy to see (using the definition of successive inradii based on linear packings) thatrC .PI .y/; m j l/ is a linear function of y. Thus, as the infimum of concave functionsis concave, (8.6) and (8.7) imply in a straightforward way that rC .P.y/; m/ is aconcave function of y, finishing the proof of Lemma 8.2.3. �

Second, observe that Lemma 8.2.3 and standard approximation by polytopesyield the following statement.

Corollary 8.2.4. Let K1; : : : ; KN , and C be convex bodies in Ed and let m be a

positive integer. Then

rC ..y1 C K1/ \ � � � \ .yN C KN /; m/

is a concave function of .y1; : : : ; yN / 2 ENd .

Finally, using Corollary 8.2.4 we get that rC .Ki .x/; m/ is a concave function ofx 2 E

d for all 1 � i � n and therefore also r.x; m/ D PniD1 rC .Ki .x/; m/ is a

concave function of x 2 Ed , finishing the proof of Theorem 8.2.2. �

8.2.3 Estimating Sums of Successive Inradii

Now, we are set for an inductive proof of Theorem 7.1.12 on the number n of tilesin the relevant inductive partition. The details are as follows. By Theorem 8.2.2 thefunction r.x; m/ D Pn

iD1 rC .Ki .x/; m/ is a concave function of x and so, Xr WDfx 2 E

d j r.x; m/ > �1g is a closed convex set in Ed . If x0 is a boundary point

of Xr , then at least one Ki .x0/ D K \ .x0 C Vi / must have an empty interiorin E

d say, intKi0 .x0/ D ; for some 1 � i0 � n. Then take the inductive partitionW1[� � �[Wi0�1[Wi0C1[� � �[Wn of Ed such that Vj � Wj for all j ¤ i0. Now, itis easy to see that if int

�K \ .x0 C Vj /

� ¤ ; for some j ¤ i0, then K\.x0CVj / D


K \ .x0 C Wj /. Thus,

nXiD1

rC .K \ .x0 C Vi /; m/ DXj ¤i0

rC�K \ .x0 C Wj /; m

�

and therefore by induction we get that the inequality r.x0; m/ � rC.K; m/ holds forall boundary points x0 of Xr . Then this fact and the concavity of r.x; m/ imply ina straightforward way that the inequality r.x; m/ � rC.K; m/ holds for all x 2 Xr

unless Xr is a closed halfspace of Ed . However, the latter case can happen only

when (each Vi , 1 � i � n contains the same halfspace and therefore) n D 1. AsTheorem 7.1.12 clearly holds for n D 1, our inductive proof of Theorem 7.1.12 iscomplete.


Theorem 7.2.2. Let K be a convex body in Ed . Let C1; : : : ; CN be k-codimensional

cylinders in Ed ; 0 < k < d such that K � SN

iD1 Ci : Then

NXiD1

crvK.Ci / � 1�dk

� :

Moreover, if K is an ellipsoid and C1; : : : ; CN are 1-codimensional cylinders in Ed

such that K � SNiD1 Ci , then

NXiD1

crvK.Ci / � 1:

8.3.1 Covering Ellipsoids by 1-Codimensional Cylinders

We prove the part of Theorem 7.2.2 on ellipsoids. In this case, every Ci can bepresented as Ci D li C Bi , where li is a line containing the origin o in E

d and Bi isa measurable set in Ei WD l?

i . Because crvK.C/ D crvT K.T C/ for every invertibleaffine map T W Ed ! E

d , we therefore may assume that K D Bd , where Bd denotesthe unit ball centered at the origin o in E

d . Recall that !d�1 WD vold�1.Bd�1/. Then

crvK.Ci / D vold�1.Bi /

!d�1

:

Consider the following (density) function on Ed ,


p.x/ WD 1=p

1 � kxk2

for kxk < 1 and p.x/ WD 0 otherwise, where k � k denotes the standard Euclideannorm in E

d . The corresponding measure on Ed we denote by �, that is, d�.x/ D

p.x/dx. Let l be a line containing o in Ed and E D l?. It follows from direct

calculations that for every z 2 E with kzk < 1,ZlCz

p.x/ dx D �:

Thus, we have

�.Bd / DZ

Bd

p.x/ dx DZ

Bd \E

ZlCz

p.x/ dx dz D � !d�1

and for every i � N ,

�.Ci / DZ

Ci

p.x/ dx DZ

Bi

Zli Cz

p.x/ dx dz D � vold�1.Bi /:

Because Bd � SNiD1 Ci , we obtain

� !d�1 D �.Bd / � �

� N[iD1

Ci

��

NXiD1

�.Ci / DNX

iD1

� vold�1.Bi /:

It implies

NXiD1

crvBd .Ci / DNX

iD1

vold�1.Bi /

!d�1

� 1:

8.3.2 Covering Convex Bodies by Cylinders

We show the general case of Theorem 7.2.2 as follows. For i � N denote NCi WDCi \ K and Ei WD H ?

i and note that

K �N[

iD1

NCi and PEiNCi D Bi \ PEi K:

Because NCi � K we also have

maxx2Ed

volk. NCi \ .x C Hi // � maxx2Ed

volk.K \ .x C Hi //:

We use the following theorem, proved by Rogers and Shephard [161] (see also [75]and Lemma 8.8 in [157]).

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes 151

Theorem 8.3.1. Let 1 � k � d � 1. Let K be a convex body in Ed and E be a

k-dimensional linear subspace of Ed . Then

maxx2Ed

vold�k.K \ .x C E?// volk.PEK/ �

d

k

!vold .K/:

We note that the reverse estimate

maxx2Ed

vold�k.K \ .x C E?// volk.PEK/ � vold .K/

is a simple application of the Fubini theorem and is correct for any measurable setK in E

d .Thus, applying Theorem 8.3.1 (and the remark after it, saying that we don’t need

convexity of NCi ) we obtain for every i � N :

crvK.Ci / D vold�k.Bi /

vold�k.PEi K/� vold�k.PEi

NCi /

vold�k.PEi K/

� vold . NCi /

maxx2Ed volk. NCi \ .x C Hi //

maxx2Ed volk.K \ .x C Hi //�dk

�vold .K/

� vold . NCi /�dk

�vold .K/

:

Using that NCi s cover K, we observe

NXiD1

crvK.Ci / � 1�dk

� ;

which completes the proof of Theorem 7.2.2.

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes

Theorem 7.3.4. If K is a spherically convex body in Sd ; d � 2, then

Svold .K/ � .d C 1/!dC1

�r.K/:

Equality holds if and only if K is a lune.

On the one hand, the proof that follows is essentially the same as the originalone published in [55]. On the other hand, it is yet somewhat more simple and itsorganization follows the relevant discussion in [2].


Lemma 8.4.1. Let � be a spherically symmetric and absolute continuous (finite)measure on E

d . Moreover, let B be a closed ball centered at the origin o in Ed , and

let T be a closed starshaped set with center o in Ed . Then

��B \ T

��E

d� � �.B/�.T/ :

Proof. Let us decompose Ed into finitely many convex cones Vi of equal measures�.Vi / having o as an apex in common in E

d . Here the measures ��B \ Vi

�are all

equal due to the spherical symmetry of �. (In other words, we start with a simpleneedle decomposition of E

d . For more details on needle decompositions see forexample, [152].) Thus, Lemma 8.4.1 will follow from the inequality

��B \ T \ Vi

��Vi

� � �.B \ Vi /�.T \ Vi / (8.8)

by summation. As the needle decomposition at hand can be made so that every Vi

gets arbitrarily close to a halfline starting at o in Ed therefore the absolute continuity

of � implies that the limit case of (8.8) becomes the following inequality on non-negative functions:

Z minfx;yg

0

f .t/dt �Z C1

0

f .t/dt �Z x

0

f .t/dt �Z y

0

f .t/dt :

This latter inequality follows from the trivial observation that minfx; yg � z � xy forany x; y 2 Œ0; z�. �

Lemma 8.4.2. Let H be a closed hemisphere with center o in Sd and let B be a

closed ball of spherical radius at most �2

and of center o in Sd . If T is a closed

starshaped set with center o in H, then

Svold .B \ T/Svold .H/ � Svold .B/Svold .T/

for the standard measure (i.e., spherical Lebesgue measure) Svold .�/ on Sd .

Proof. Let Ed be the hyperplane tangent to the unit sphere Sd at the point o in EdC1.

Then the central projection of Sd onto Ed (keeping the point o fixed) combined with

Lemma 8.4.1 implies Lemma 8.4.2. �For the purpose of the next statement let BŒx; "� denote the closed ball of spherical

radius " � �2

centered at the point x in Sd . Furthermore, for any subset X of Sd let

X" WD [x2XBŒx; "� be called the "-neighbourhood of X in Sd .

Lemma 8.4.3. Let X be a closed subset of Sd not lying on an open hemisphere ofS

d . Then for any " � �2

the following inequality holds:

Svold .X"/ � Svold . OX"/ ;

where OX is a pair of antipodal points of Sd .

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes 153

Fig. 8.1 The point xi and itsVoronoi cell Vi and thehemisphere Hi centered at xi

Proof. It is sufficient to prove Lemma 8.4.3 for finite X say, for X WDfx1; x2; : : : ; xmg � S

d . Take the (nearest point) Voronoi tiling of Sd generated

by X with Vi standing for the Voronoi cell assigned to the point xi , 1 � i � m. LetHi be the closed hemisphere of Sd centered at xi , 1 � i � m. (See Fig. 8.1.) As, byassumption, X does not lie on an open hemisphere of Sd therefore, Vi � Hi holdsfor all 1 � i � m. Thus, Lemma 8.4.2 yields that

Svold .BŒxi ; "� \ Vi /Svold .Hi / � Svold .BŒxi ; "�/Svold .Vi / ;

or equivalently

Svold .X" \ Vi /Svold .Sd /

2� Svold .BŒxi ; "�/Svold .Vi / (8.9)

holds for all 1 � i � m. AsPm

iD1 Svold .X" \ Vi / D Svold .X"/ andPmiD1 Svold .Vi / D Svold .Sd / therefore (8.9) implies in a straightforward way that

Svold .X"/ � 2Svold .BŒx; "�/

holds for any x 2 Sd , finishing the proof of Lemma 8.4.3. �

Lemma 8.4.4. Let � be a spherically symmetric and absolute continuous (finite)measure on E

d . Moreover, let K be a convex body in Ed with the inscribed (closed)

ball B centered at the origin o in Ed . Then

�.K/ � �.P/ ;

where P is a plank of Ed having B as an inscribed ball.


Proof. As the measure � is absolute continuous therefore representing it as anintegral, it is sufficient to prove the inequality

�.K \ S/ � �.P \ S/ (8.10)

for any .d � 1/-dimensional sphere S of Ed with center o and with the proper

spherical Lebesgue measure � on it. Let r (resp., r) be the radius of S (resp., B). Ifr � r , then the inequality (8.10) is obvious. So, we are left with the case r < r . Themore exact details are as follows.

Let X 0 WD �bd K

� \ B (resp., OX 0 WD �bd P

� \ B) and X WD rrX 0 � S (resp.,

OX WD rr

OX 0 � S ). Clearly, X is not contained in an open hemisphere of S and of

course, OX is a pair of antipodal points on S . Now, let " WD arccos rr

< �2

and forany Y � S let Y" denote the set of those points of S whose angular distance fromY is at most " in S . It is straightforward to see that

X" \ �.int K/ \ S

� D ; (8.11)

and

OX" \ �.int P/ \ S

� D ; and OX" [ �P \ S

� D S : (8.12)

By Lemma 8.4.3

�.X"/ � �. OX"/ : (8.13)

Thus, (8.11)–(8.13) imply in a straightforward way that

�.K \ S/ � �.S/ � �.X"/ � �.S/ � �. OX"/ D �.P \ S/ ;

finishing the proof of (8.10) as well as Lemma 8.4.4. �Finally, let Ed be the hyperplane of EdC1 that is tangent to the unit sphere S

d

touching it at the center of the inscribed ball of the convex body K in Sd . Then

the central projection of Sd (from its center) onto Ed combined with Lemma 8.4.4

proves Theorem 7.3.4.As a last note we mention that the above proof implies the following stronger

(truncated) version of Theorem 7.3.4.

Theorem 8.4.5. Let K be a convex body in Sd with the inscribed (closed) ball B

centered at o 2 Sd . Furthermore, let B0 be any (closed) ball of Sd centered at o.

Then

Svold�K \ B0� � Svold

�L \ B0� ;

where L is a lune of Sd having B as an inscribed ball.

8.6 Proof of Theorem 7.4.6: Lower Bounds for Ball-Polyhedra 155

8.5 Proof of Theorem 7.4.2: Partial Coveringsin Euclidean 3-Space

Theorem 7.4.2. Let w1; w2; : : : ; wn be positive real numbers satisfying the inequal-ity w1 C w2 C � � � C wn < 2. Then the union of the planks P1; P2; : : : ; Pn ofwidth w1; w2; : : : ; wn in E

3 covers the largest volume subset of B3 if and only ifP1 [ P2 [ � � � [ Pn is a plank of width w1 C w2 C � � � C wn with o as a center ofsymmetry.

Let P1; P2; : : : ; Pn be an arbitrary family of planks of width w1; w2; : : : ; wn in E3

and let P be a plank of width w1 C w2 C � � � C wn with o as a center of symmetry.Moreover, let S.x/ denote the sphere of radius x centered at o. Now, recall the well-known fact that if P.y/ is a plank of width y whose both boundary planes intersectS.x/, then sarea.S.x/ \ P.y// D 2�xy, where sarea. : / refers to the surface areameasure on S.x/. This implies in a straightforward way that

sareaŒ.P1 [ P2 [ � � � [ Pn/ \ S.x/� � sarea.P \ S.x//;

and so,

vol3..P1 [ P2 [ � � � [ Pn/ \ B3/ DZ 1

0

sareaŒ.P1 [ P2 [ � � � [ Pn/ \ S.x/� dx

�Z 1

0

sarea.P \ S.x// dx D vol3.P \ B3/;

finishing the proof of the “if” part of Theorem 7.4.2. Actually, a closer look of theabove argument gives a proof of the “only if” part as well.

8.6 Proof of Theorem 7.4.6: Lower Boundsfor Ball-Polyhedra

Theorem 7.4.6. Let BŒX� � Ed be a ball-polyhedron of minimal width x with

1 � x < 2. Thenvold .BŒX�/ � vold .K2�x;d

BL / C svold�1.bd.K2�x;d

BL //.x � 1/ C vold .Bd /.x � 1/d .

Recall that if X is a finite set lying in the interior of a unit ball in Ed , then we can

talk about its spindle convex hull convs.X/, which is simply the intersection of all(closed) unit balls of Ed that contain X (for more details see [59]). The followingstatement can be obtained by combining Corollary 3.4 of [59] and Proposition 1of [58].


Lemma 8.6.1. Let X be a finite set lying in the interior of a unit ball in Ed . Then

(i) convs.X/ D B�BŒX�

�and therefore BŒX� D B

�convs.X/

�,

(ii) The Minkowski sum BŒX� C convs.X/ is a convex body of constant width 2 inE

d and so, w.BŒX�/ C diam�convs.X/

� D 2, where diam. : / stands for thediameter of the corresponding set in E

d .

By part .ii/ of Lemma 8.6.1, diam�convs.X/

� � 2 � x. This implies, via aclassical theorem of convexity (see, e.g., [65]), the existence of a convex body Lof constant width .2 � x/ in E

d with convs.X/ � L. Hence, using part .i/ ofLemma 8.6.1, we get that BŒL� � BŒX� D B

�convs.X/

�. Finally, notice that as L is

a convex body of constant width .2 � x/ therefore BŒL� is, in fact, the outer-paralleldomain of L having radius .x � 1/ (i.e., BŒL� is the union of all d -dimensional(closed) balls of radii .x � 1/ in E

d that are centered at the points of L). Thus,

vold .BŒX�/ � vold .BŒL�/ D vold .L/Csvold�1.bd.L//.x �1/Cvold .Bd /.x �1/d :

The above inequality together with the following obvious ones

vold .L/ � vold .K2�x;dBL / and svold�1.bd.L// � svold�1.bd.K

2�x;d

BL //

implies Theorem 7.4.6 in a straightforward way.

[fields institute monographs] lectures on sphere arrangements – the discrete geometric side volume...

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