Convex Bodies with Minimal Mean Width A

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A Giannopoulos VD Milman and M Rudelson Department of Mathematics University of Crete Iraklion Greece School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel Department of Mathematics University of Missouri Columbia MO 6 ID: 29176 Download Pdf

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Convex Bodies with Minimal Mean Width A

A Giannopoulos VD Milman and M Rudelson Department of Mathematics University of Crete Iraklion Greece School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel Department of Mathematics University of Missouri Columbia MO 6

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Convex Bodies with Minimal Mean Width A




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Convex Bodies with Minimal Mean Width A.A. Giannopoulos , V.D. Milman , and M. Rudelson Department of Mathematics, University of Crete, Iraklion, Greece School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel Department of Mathematics, University of Missouri, Columbia, MO 65211, USA 1 Introduction Let be a convex body in , and TK SL be the family of its positions . In [GM] it was shown that for many natural functionals of the form 7 TK , T SL the solution of the problem min TK SL is isotropic with respect to an appropriate measure depending on . The

purpose of this note is to provide applications of this point of view in the case of the mean width functional 7 TK ) under various constraints. Recall that the width of in the direction of is defined by K,u ) = ) + ), where ) = max x,y is the support function of . The width function K, ) is translation invariant, therefore we may assume that int( ). The mean width of is given by ) = K,u du ) = 2 du where is the rotationally invariant probability measure on the unit sphere We say that has minimal mean width if TK ) for every SL ). The following isotropic characterization of the minimal

mean width position was proved in [GM]: Fact. A convex body in has minimal mean width if and only if u, du ) = for every . Moreover, if SL and UK has minimal mean width, we must have Research of the second named author partially supported by the Israel Science Foundation founded by the Academy of Sciences and Humanities. Research of the third named author was supported in part by NSF Grant DMS-9706835.
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2 A.A. Giannopoulos et al. Our first result is an application of this fact to a “reverse Urysohn in- equality” problem: The classical Urysohn inequality states that / /n

where is the volume of the Euclidean unit ball , with equality if and only if is a ball. A natural question is to ask for which bodies := max =1 min SL TK is attained, and what is the precise order of growth of as . Ex- amples such as the regular simplex or the cross-polytope show that log( + 1). On the other hand, it is known that every symmetric con- vex body in has an image TK with TK = 1 for which TK log[ ,` ) + 1] where = ( k·k ) and denotes the Banach-Mazur distance. This statement follows from an inequality of Pisier [Pi], combined with work of Lewis [L], Figiel and Tomczak-Jaegermann

[FT]. John’s theorem [J] implies that min SL TK log( + 1) for every symmetric convex body with = 1, and a simple argument based on the difference body and the Rogers-Shephard inequality [RS] shows that the same holds true without the symmetry assumption. Therefore, log( + 1) log( + 1) Here, we shall give a precise estimate for the minimal mean width of zonoids (this is the class of symmetric convex bodies which can be approximated by Minkowski sums of line segments in the Hausdorff sense): Theorem A. Let be a zonoid in with volume = 1 . Then, min SL TZ ) = where = [ 2] For our

second application, we consider the class of origin symmetric convex bodies in . Every symmetric body induces a norm min 0 : λK on , and we write for the normed space ( k· ). The polar body of is defined by = max | x,y 〉| ), and will be denoted by . Whenever we write (1 /a |≤k , we assume that a,b are the smallest positive numbers for which this inequality holds true for every We consider the average ) = dx
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Convex Bodies with Minimal Mean Width 3 of the norm k·k on , and define ) = ). Thus, ) is half the mean width of . We will say that has

minimal if TK for every SL ). Equivalently, if has minimal mean width. Our purpose is to show that if has minimal , then the volume radius of is bounded by a function of and . Actually, it is of the order of b/M The precise formulation is as follows: Theorem B. Let be a symmetric convex body in with minimal such that (1 /a |≤k . Then, (1 /b /n log where c> is an absolute constant. Our last result concerns optimization of the width functional under a different condition. We say that an -dimensional symmetric convex body is in the Gauss-John position if the minimum of the functional

TK under the constraint TK is attained for . That is, has minimal mean width under the condition TK (it minimizes under the condition TK 1). We can consider this optimization problem only for positive self-adjoint operators . Since the norm of should be bounded to guarantee that TK and the norm of should be bounded as well, there exists for which the minimum is attained. Denote by the standard Gaussian measure in Then, we have the following decomposition. Theorem C. Let be in the Gauss-John position. Then there exist: +1) , contact points ,...,x ∂K and numbers ,...,c such that =1 = 1 and

d ) = d =1 The Gauss-John position is not equivalent to the classical John position. Examples show that, when is in the Gauss-John position, the distance between and the John ellipsoid may be of order n/ log 2 Reverse Urysohn Inequality for Zonoids The proof of Theorem A will make use of a characterization of the minimal surface position, which was given by Petty [Pe] (see also [GP]): Recall that the area measure of a convex body is defined on and corresponds
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4 A.A. Giannopoulos et al. to the usual surface measure on via the Gauss map: For every Borel , we have ) = bd( )

: the outer normal to at is in where is the ( 1)-dimensional surface measure on . If ) is the surface area of , we obviously have ) = ). We say that has minimal surface area if TK ) for every SL ). With these definitions, we have: 2.1 Theorem. A convex body in has minimal surface area if and only if u, du ) = for every . Moreover, if SL and UK has minimal surface area, we must have Recall also the definition of the projection body ΠK of : it is the sym- metric convex body whose support function is defined by ΠK ) = where ) is the orthogonal projection of onto . It

is known that is a zonoid in if and only if there exists a convex body in such that ΠK . By the formula for the area of projections, this can be written in the form ) = | x,u 〉| du Then, the characterization of the minimal mean width position and Theorem 2.1 imply the following: 2.2 Lemma. Let ΠK be a zonoid. Then, has minimal mean width if and only if has minimal surface area. Proof. The proof (modulo the characterization of the minimal mean width position) may be found in [Pe]: By Cauchy’s surface area formula, ) = n d If is a spherical harmonic of degree 2, the Funk-Hecke

formula shows that | u, 〉| du ) = for all u, , where is a constant depending only on the dimension. Therefore, du ) = | u, 〉| du d d
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Convex Bodies with Minimal Mean Width 5 Since 7 u, is homogeneous of degree 2, this implies u, du ) = u, du for every . The characterizations of the minimal mean width and the minimal surface area positions make it clear that has minimal mean width if and only if has minimal surface area. Our next lemma is a well-known fact, proved by K. Ball [B]: 2.3 Lemma Let be unit vectors in and be positive numbers satisfying =1 If =1 ,u for

some , then | =1 We apply this result to the projection body of a convex body with minimal surface area. 2.4 Lemma If has minimal surface area, then ΠK /n Proof. We may assume that is a polytope with facets and normals = 1 ,...,m . Then, Theorem 2.1 is equivalent to the statement =1 where /A ) (see [GP]). On the other hand, ΠK =1 ,u We now apply Lemma 2.3 for ΠK , with ΠK | =1 Remark. In the previous argument, equality can hold only if ( is an orthonormal basis of (see [Ba]). This means that if is a polytope then equality in Lemma 2.3 can hold only if is a cube.
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6 A.A. Giannopoulos et al. Proof of Theorem A. Let be a zonoid with minimal mean width and volume = 1. By Lemma 2.2, is the projection body ΠK of some convex body with minimal surface area. We have ) = 2 du ) = 2 du ) = n By Lemma 2.4, the area of is bounded by /n . We have equality when is a cube, and this corresponds to the case . Therefore, ) = Remark. Urysohn’s inequality and Theorem A show that if is a zonoid with = 1, then πe min SL TZ n, where , 1 as 3 Volume Ratio of Symmetric Convex Bodies with Minimal For the proof of Theorem B we will need the following fact which

was proved in [GM]: 3.1 Theorem. Let be a symmetric convex body in with minimal Then, for every (0 1) there exists a [(1 -dimensional subspace of such that |≤k , x E, (3 1) where M log( M , and c> is an absolute constant. Actually, the proof of Theorem 3.1 shows that the statement holds true for a random [(1 ]-dimensional subspace of . One can assume that for every log we have the result with probability greater than (this formulation is correct when , where is absolute). This assumption on the measure of subspaces satisfying (3.1) implies that there is an increasing sequence of

subspaces ... , where = [ log ] and dim , so that (3.1) holds for each with k/n ). We will also need the following
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Convex Bodies with Minimal Mean Width 7 3.2 Lemma. Let be a symmetric convex body in , such that (1 /a | . If is a -dimensional subspace of , then Ca where C > is an absolute constant. Proof. Let be a -dimensional subspace of . Replacing by (1 /a , we may assume that = 1, so . Using Brunn’s theorem we see that dy ≤| || ≤| || This shows that || Proof of Theorem B. We first observe that ab Cn log n. Indeed, let be such that and let be a standard

normal variable. Then cb γe k Similarly, ca Multiplying these inequalities, we obtain ab CE Cn log n. The last inequality follows from the fact that is minimal for , and Pisier’s inequality [Pi]. Assume now that = 1. Let = [log ]. For = 1 ,...,t put [(1 /s ] and let be a subspace from our flag. Then by Theorem 3.1 we have (1 /a |≤k ≤| on , where (( /n log log =: Now, Lemma 3.2 shows that Ca Cn log n/ log
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8 A.A. Giannopoulos et al. and +1 +1 Ca +1 Cc +1 for all = 1 ,...,t . Since ), we have | Hence, multiplying the inequalities above, we get Cc )) =2 2(

+1 By the definition of =2 2( +1 exp cn =2 log cn therefore (1 /b /n The left hand side inequality is an immediate consequence of H¨older’s in- equality: (1 /b /n dx /n 4 Gauss-John Position We prove Theorem C. Consider the following optimization problem: ) = d min (4 1) under the constraint ) = Tx 0 for K. Assume that the body is in the Gauss-John position, namely the minimum in (4.1) is attained for . Let be the set of the contact points of ∂K ∂S . First we apply an argument of John [J] to show that we
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Convex Bodies with Minimal Mean Width 9 can consider

only finitely many constraints. Since the paper [J] is not easily available, we shall sketch the argument. Let be a self-adjoint operator and let sT . We shall prove that if ds =0 for every , then ds =0 Indeed, assume that = sup ds =0 Let be an -neighborhood of dist( x,W < . There exists an ε> 0 such that ds =0 for every . So, there exists 0 such that for any 0 < s < s and any On the other hand, if then ≤| Here, −| ≤k (1 + and sup ) = sup since is compact. Thus for a sufficiently small we have 0 for all . So, since is the solution of the minimization problem

(4.1), ds =0 Since ds =0 ,T and ds =0 ,T , this means that the vector ) cannot be separated from the set { by a hyperplane. By Carath´eodory’s theorem, there exist +1) contact points ...x and numbers ... 0 such that ) = =1 ) = =1 (4 2)
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10 A.A. Giannopoulos et al. Now we have to calculate ). We have ) = (2 n/ −| dx = (2 n/ det −| Tx dx, so, ) = (2 n/ −| dx (2 n/ −| xdx ·k d Combining it with (4.2) we obtain ·k d ) + =1 = 0 Taking the trace, we get Tr ·k d d ·k d dr dm +2 dr dm d Finally, putting d ), we obtain the decomposition d ) = d =1 where =1 = 1.

This completes the proof of Theorem C. We proceed to compare with the John ellipsoid in the Gauss-John position: Proposition. Let be a symmetric convex body in which is in the Gauss- John position. Then, (i) (2 /
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Convex Bodies with Minimal Mean Width 11 (ii) It may happen that log is not contained in Proof. (i) Let be an operator which puts into the maximal volume position. Then min . From the other side, if there exists such that (2 / then d | x,y/ 〉| d (2 / >n. (ii) Let + [ ,e ]. Let be a positive self-adjoint operator such that TK is in the Gauss-John position. We

first prove that is a diagonal operator. Let ) be the group generated by the operators , i = 1 ,...,n and let be the uniform measure on . Notice that for every . Then, TK d ) = Ux TU d dm U dm d Put Udm ) = diag( We claim that diag( Indeed, since for any diag( and diag( ,Te the claim follows from the fact that for any θ,T 〉· θ,T Let = diag( ). Since , we have ) = TK d Wx d d ) = [Notice that since TK SK TU dm so the restrictions of the optimization problem (4.1) are satisfied.]
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12 A.A. Giannopoulos et al. Let now ) be the group generated by the

operators ij for i,j = 1 ,...,n , i . Arguing the same way we can show that there exist a,b> 0 such that ), where =1 be Since the vertices of are contact points, = 1 We have = max =1 ,b Denote =1 and let ) = ( b/a ·k . Then, ) = d dx dx d ) + d where ) = (1 du . We have to show that log . We may assume that c/n . Putting = (1 and differentiating, we get after some calculations db ) = d Since c/n and Cn with probability at least 1/2, we have >c with probability 1/2, for some absolute constant c> 0. So, db Cb exp( cn which is positive when log n/n Remark. The dual problem ) = sup TK x,y d

max under the constraint ) = Tx 0 for is very different. The examples suggest that the matrix for which the maximum is attained may be singular.
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Convex Bodies with Minimal Mean Width 13 References [B] Ball K.M. (1991) Shadows of convex bodies. Trans. Amer. Math. Soc. 327:891–901 [Ba] Barthe F. (1998) On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134:335–361 [BM] Bourgain J., Milman V.D. (1987) New volume ratio properties for convex symmetric bodies in . Invent. Math. 88:319–340 [FT] Figiel T., Tomczak-Jaegermann N. (1979) Projections onto Hilbertian

sub- spaces of Banach spaces. Israel J. Math. 33:155–171 [GM] Giannopoulos A.A., Milman V.D. Extremal problems and isotropic posi- tions of convex bodies. Israel J. Math., to appear [GP] Giannopoulos A.A., Papadimitrakis M. (1999) Isotropic surface area mea- sures. Mathematika 46:1–13 [J] John F. (1948) Extremum problems with inequalities as subsidiary condi- tions. Courant Anniversary Volume, Interscience, New York, 187–204 [L] Lewis D.R. (1979) Ellipsoids defined by Banach ideal norms. Mathematika 26:18–29 [MS] Milman V.D., Schechtman G. (1986) Asymptotic theory of finite dimen-

sional normed spaces. Lecture Notes in Mathematics, 1200, Springer, Berlin [Pe] Petty C.M. (1961) Surface area of a convex body under affine transforma- tions. Proc. Amer. Math. Soc. 12:824–828 [Pi] Pisier G. (1982) Holomorphic semi-groups and the geometry of Banach spaces. Ann. of Math. 115:375–392 [RS] Rogers C.A., Shephard G. (1957) The difference body of a convex body. Arch. Math. 8:220–233