a characterization of parallelepipeds related to

8/6/2019 A Characterization of Parallelepipeds Related To

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A characterization of parallelepipeds related to

weak derivatives

Heinz Weisshaupt

University of Duesseldorf, Mathematical Institute

Universitaetsstrasse 1, D-40225 Duesseldorf, Germany

email: [email protected]

July 2, 2003

Abstract

We characterize in this paper parallelepipeds in Rm within the familyof all convex bodies by a property of special measures on its boundary.We show that these measures are related to weak derivatives (in the senseof [5] and [8]) of convex-valued functions. The results can be applied (see[9]) to derive a generalization of a theorem of Lehmann (see [4]) on thecomparison of uniform location experiments.

Keywords: parallelepipeds, convex bodies, characterization, weak derivatives,convex-valued functions

AMS Subject Classification: 52A20, 52A38, 28A15, 26E25

1 Introduction

The investigation of special convex bodies and especially their characterizationis a main topic in classical convex geometry (see [2] chapter 1.11). There existseveral characterizations of parallelepipeds, simplexes and ellipsoids within thefamily of convex bodies (see [3], [1], [7] and [2]). In section 2 of this paper wegive a new characterization of parallelepipeds by a property of certain measureson the boundary of convex sets. In section 3 we define measure valued weakderivatives of convex valued mappings. We show that the measures used tocharacterize parallelepipeds in section 2 are related to the derivatives of convex

valued mappings given by shifts of the parallelepiped (convex body) in the di-rections of its 1-dimensional edges. We note that the results of section 3 can beused to prove a theorem on the comparison of uniform location experiments [9]which generalizes Theorem 3.1 of [4].

We denote by N = {1, 2, . . . } the set of positive integers. By Sm1 we denotethe unit sphere in Rm, by ., . we denote the euclidean scalar product on Rm

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and by .2 we denote the norm associated with ., .. By V we denote the

closure of a set V, by 1B we denote the indicator function of a set B and by xwe denote the Dirac measure at a point x. Given a countable set M we denoteby M the counting measure on M, i.e. M(A) =

mM m(A). Given two

measures and on Rm we denote by the convolution of and . If isabsolutely continuous with respect to we write


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Proof: If C is a parallelepiped we just have to choose wi such that wi

equals the length of the one dimensional edge of C which is parallel to wi,R = {1, 0, 1} and r0 = 0 independent of j. Then

[Cwj {rwj |rR}](B) = 0 for all Borel measurable sets B (C + r0 wj)

which proves the only if part. The only if part is the consequence of the lemmas1, 2 and 3 which we state and prove below. 2

Proposition 1 Given a convex body C, a vector w = 0, a real number 1 anda point x C + 1w. If for arbitrary hyper planes H

Hw H x[C + 1w]

then there exists one and only one 2 = 1 such that x C + 2w.

Proof. We show that x + Rw int(C + 1w) = . Otherwise there wouldexist a hyperplane H which separates x + Rw and C + 1w. This hyperplanewould fulfill H x[C+1w] and Hw which contradicts our hypothesis. Thusthere exists y (x +Rw) int(C + 1w). By compactness and convexity of Cwe get that (x +Rw) (C + 1w) = (y +Rw) (C + 1w) consists of exactlytwo points. One of them is x the other one is x + w for some R. Thus for2 = 1 we have x C + 2w if and only if 2 = 1 . 2

Proposition 2 Given a vectorw = 0. Suppose that the intersection of a convexbody C and its translate C + w (with = 0) is again a convex body, i.e.,int(C (C + w)) = . Then

oC({x | x C (C + w) s.t. [Hw H xC]}) = 0 (1)

and

oC+w({x | x C (C + w) s.t. [Hw H xC]}) = 0 (2)

Proof. We prove only (1), since the proof of (2) is completely analogous.Let

X = {x | x C (C + w) s.t. [Hw H xC]}.

We have to show that oC(X) = 0. We show first that x X implies that

(x + Rw) X = x. (3)

We prove (3) indirect: Suppose that y = x such that y (x + Rw) X.Then y x = w with = 0 and we obtain from the definition of X that

y C + 0, y C + w, y C + w, y C + ( + )w .

Since 0 = = = 0 three of the four numbers 0, , , + must be pairwisedifferent which contradicts together with the fact that y X Proposition 1. So(3) has been proved.

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Let Z := {z Rm | inf{x|zxRw} x2 = 1} and let S := {z Z | z2 =

1}. We suppose without loss of generality that 0 int(C (C + w)). ThenX Rw = and thus X \ Rw = X. We denote by pr : X Z the projectionof X along the rays from the origin to Z. By the fact that X \ Rw = X themapping pr is well defined. We identify Z = S + Rw with S R and denoteby := oS 1 the measure on Z which is the product of the surface areameasure oS on S and the Lebesgue measure 1 on R. Note that by (3) we getfor x, x X that

x = x implies that for (s, r), (s, r) S R with(s, r) = pr(x) and (s, r) = pr(x) we have s = s.

(4)

Then (pr(X)) =S

R

1pr(X) d1 doS = 0 by Fubinis theorem and (4).Since oC is absolutely continuous with respect to pr we obtain that oC(X) = 0which proves the proposition. 2

We state now two further propositions without proofs.

Proposition 3 Given a convex body C and a vector w Rm \ {0}, then thereexist two numbers 1 and 2 R such that C (C+ i) = and int(C (C+i)) = . Further there exists a hyperplane F such that FC (C + i).

Proposition 4 Given a convex body C and a vector w = 0 then

oC{x C s.t. [Hw H xC]} > 0

Lemma 1 Given a countable set R R, an element 0 R, a convex body C,and a vector w Rm \ {0}. Suppose that

[Cw R](B) = 0 for all Borel measurable sets B (C + 0w) . (5)

Then the following conclusion holds:There exists a hyperplane Fw such that

x C [H xC s.t. Hw or H xC s.t. HF]

Proof. We suppose without loss of generality that 0 = 0. Let

Y = {x C s.t. [Hw H xC]}.

Then oC(Y) > 0 by Proposition 4. Since there exists a version ofdCwdoC

(y) such

that 0 =dCwdoC

(y) for all y Y we get that

Cw |Y= 0. (6)

By Proposition 2 we obtain that if C (C + w) is a convex body and 0 = then

o(C+w)(Y) = o(C+w)(Y (C + w)) =

o(C+w)({x | x C (C + w) s.t. [Hw H xC]}) = 0 .(7)

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Since Cw w = C+ww > Cw |Y . (10)

Since oC |Y 0 and some convex setCw w with dim(Cw) = m 1.

Proof: Of course there exist exactly two hyperplanes G1 and G2 parallel toF which support C.

Let y int(C). Then there exist reals y < y such that y + yw, y + yw C. Since y + yw and y + yw can not possess a hyperplane of support whichis parallel w they must by hypotheses possess a hyperplane of support parallelF. So we get, probably by an interchange of G1 and G2, that

y + yw G1 C and y + yw G

2 C. (11)

By (11) it is clear that = y y is independent of y and G2 = G1 + w.Let D1 = G1 C and let D2 = G2 C. Note that

D1 = G1 C = {y + yw | y int(C)}

D2 = G1 C = {y + yw | y int(C)}.

(12)

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Thus D1 and D2 are compact convex sets of dimension m 1 with

D2 = D1 + w. (13)

So we get by (11), (13), (12) and the compactness of C that

int(C) conv(D1 D2) = D1 + [0, ]w C = C. (14)

Since D1 + [0, ] is compact we get from (14) that D1 + [0, ]w = C. Thiscompletes the proof if we let Cw = D

1. 2

Lemma 3 LetC = Ci+[0, i]wi for(w1, . . . , wm) linearly independent vectorsin Rm and Ci wi convex compact sets with dim(Ci) = m 1. Then C is aparallelepiped with 1-dimensional edges parallel with wi.

Proof. Note that i {1, , m}

x C (x Ci or x Ci + i or [H x(C) Hwi]). (15)

From (15) we derive

x C i s.t. [x Ci or x Ci + i]. (16)

Indirect: Otherwise by (15) H x(C) fulfills Hwi i {1, . . . , m}. Since{wi | i = 1, . . . , m} spans by hypotheses Rm no hyperplane can be parallel withall wi. Thus we obtained a contradiction and (16) is proved.

Let pri be the projection along H onto Rwi, i.e., let pri(x) = y Rwi s.t.x y H. Then C

mn=1pr

1i (pri(C)) =: D and D is a parallelepiped which

can for all i = 1, . . . , m be written as D = Di + [0, i]wi with Di Ci.So the lemma is proved if we show that C = D. We proceed indirect: Since

C is a convex body C = D implies that there exists x C with x D .But by (16) there exists an i {1, . . . m} such that x Ci Di D orx Ci + iwi Di + iwi D which contradicts the fact that x D andthus completes the proof of the lemma.2

3 Weak differentiation and a second Version of

the Characterization Theorem

Definition 3 We denote by Cc the space of continuous real valued functionswith compact support onRm. We remark that a signed measure is determinedby the integrals

d with Cc. Thus we can define the derivative of a set

valued function as follows: Let D : [0, ) P(Rm). If there exists a measure such that for any Cc

d = limh0

1D(h) 1D(0)h

d

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then we say that is the derivative from the right of D(.) at 0. (We also say

that is the weak limit limh01D(h)1D(0)

h .)

Remark 2 The measure Cw can also be described as follows: Let H be thehyperplane perpendicular to w. Let H denote the m 1-dimensional Lebesguemeasure on H. Let

Y+ := {x C | (x), w > 0}

andY := {x C | (x), w < 0}

Let pr+ : Y+ H respectively pr : Y

H be the orthogonal projectionsonto H. Then we have for any Lebesgue measurable B Rm that

xP

w2 (pr1+ (x))dH(x) xP

w2 (pr1 (x))dH(x) =

dCw

We note further that pr+(Y+) = pr(Y

) and let P := pr+(Y+) = pr(Y

).These facts can be established easily for Polytopes or convex bodies with differ-entiable boundary and then extend without difficulties to arbitrary convex bodies.

We calculate now the derivative from the right at 0 of the special set valuedfunction D(h) = C + h, for a convex body C. (Compare with [8] Example 1.)

Proposition 5 Let C be a convex body then the derivative from the right ofh (C + h w) equals Cw , i.e.

d

C

w = limh0

1(C+hw) 1C

h d

Proof: We use the notation of Remark 2. Further we denote by 1 theLebesgue measure on R. Then we have for Borel measurable sets H H andR R that

(H +

R) = 1 H(R H) (17)

We obtain that

limh0

Rm

1(C+hw) 1C

hd =

limh0

,xRH

(x + w

w2)

1(C+hw) 1Ch

(x + w

w2) d[1 H](, x) =

limh0

xH

R

(x + ww2

) 1(C+hw) 1Ch

(x + ww2

) d1() dH(x) =

xP

limh0

1h

[0,hw2]

(pr1+ (x) + w

w2) d1()

dH(x)

xP

limh0

1h

[0,hw2]

(pr1 (x) + w

w2) d1()

dH(x) =

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xP

w2 (pr1+ (x))dH(x)

xP

w2 (pr1 (x))dH(x) =

dCw

The first equality sign holds by (17), the second by Fubinis Theorem, thefourth by the main theorem of calculus and the last equality sign follows byRemark 2. 2

If we denote by U(C+h w) the uniform probability distribution on C+h wand note that

1(C+hw) 1C

hd = (C)

d

U(C + h w) U(C)

h

then we can reformulate Proposition 5 as follows:

Corollary 1 LetC be a convex body. Then the weak limit (C)limh0 U(C+hw)U(C)

h

equals Cw , i.e.,

(C) limh0

d

U(C + h w) U(C)

h=

dCw for all Cc. (18)

With Corollary 1 we can restate Theorem 1 as follows:

Theorem 2 A convex body C Rm is a parallelepiped with one dimensionaledges parallel {w1, . . . , wm} if and only if there exists a linearly independent set{w1, . . . , wm} of vectors wj R

m such that for any j {1, . . . , m} there existsa countable set R R and an r0 R such that

[(C) limh0

U(C + h w) U(C)h

{rwj |rR}](B) = 0.

for any Borel measurable set B C + r0 wj.

References

[1] Buchman, E.; Valentine, F.A. A characterization of the parallelepided inEn Pac. J. Math. 35, 53-57 (1970).

[2] Gruber, P.M.(ed.); Wills, J.M.(ed.) Handbook of convex geometry. VolumeA. Amsterdam: North-Holland (1993).

[3] Guggenheimer, H.; Lutwak, E. A characterization of the n-dimensionalparallelotope. Am. Math. Mon. 83, 475-478 (1976).

[4] Lehmann, E.L. Comparing location experiments. Ann. Stat. 16, No.2,521-533 (1988).

[5] Pflug, Georg Optimisation of Stochastic Models. Kluwer Academic Pub-lishers, Boston, 1996.

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[6] Phelps, Robert Convex Functions, Monotone Operators and Differentiabil-

ity. 2nd Edition, Springer Verlag, Berlin Heidelberg (1993)[7] Valentine, Frederick A. Convex sets (McGraw-Hill Series in Higher Math-

ematics) New York-San Francisco-Toronto-London: McGraw-Hill BookCompany. (1964).

[8] Weisshaupt, Heinz A measure-valued approach to convex set-valued dy-namics. Set-Valued Anal. 9, No.4, 337-373 (2001)

[9] Weisshaupt, Heinz A generalization of Lehmanns Theorem on the com-parison of uniform location experimentsEURANDOM Report series Report 2003-014http://www.eurandom.tue.nl/publications.htm

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