On the approximation of a polytope by its dual L p -centroid bodies

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1 On the aroximation of a olytoe by its dual L -centroid bodies Grigoris Paouris and Elisabeth M. Werner Abstract We show that the rate of convergence on the aroximation of volumes of a convex symmetric olytoe P R n by its dual L -centroid bodies is indeendent of the geometry of P. In articular we show that if P has volume, «P = n. log P We rovide an alication to the aroximation of olytoes by uniformly convex sets. Introduction Let K be a convex body in R n of volume and, for δ,, let K δ be the convex floating body of K []. It is the intersection of all halfsaces H + whose defining hyerlanes H cut off a set of volume δ from K. Note that K δ converges to K in the Hausdorff metric as δ. C. Schütt and the second name author showed an exact formula for the convergence of volumes [], δ K K δ = as K, δ n+ which involves the affine surface area of K, as K. The same henomenon and similar formulas has been observed for other tyes of aroximation using instead of floating bodies, convolution bodies [], illumination bodies [7] or Santaló bodies [8]. We refer to e.g. [], [4]-[9], []-[7], [3]-[6], [8]-[3] for further details, extensions and alications. Another family of bodies that aroximate a given Keywords: centroid bodies, floating bodies, olytoes, L Brunn Minkowski theory. Mathematics Subject Classification: 5A, 53A5 Partially suorted by an NSF grant DMS-965 Partially suorted by an NSF grant, a FRG-NSF grant and a BSF grant

2 convex body K are the L -centroid bodies of K introduced by Lutwak and Zhang [7]. For a symmetric convex body K of volume in R n and n, the L -centroid body Z K is the convex body that has suort function h ZKθ = K x, θ dx, θ S. Note that Z K converges to K in the Hausdorff metric as. It has been shown in [9] that the family of L -centroid bodies is isomorhic to the family of the floating bodies: K δ is isomorhic to Z log K. However, it was roved in [9] δ that in the case of C+ bodies, the convergence of volume of the L -centroid bodies is indeendent of the geometry of K: For any symmetric convex body in R n of volume that is C+ i.e. K has C boundary with everywhere strictly ositive Gaussian curvature, log K K = nn + K. In this work we show that the same henomenon occurs also in the case of olytoes. We show the following Theorem.. Let K be a symmetric olytoe of volume in R n. Then log K K = n K. As an alication of this result we get bounds for the aroximation of a olytoe by a uniformly convex body with resect to the symmetric difference metric: Theorem.. Let P be a symmetric olytoe in R n. Then there exists = P such that for every, there exists a -uniformly convex body K such that d s P, K n P log, where d s is the symmetric difference metric. The statements and roofs are for symmetric convex bodies only. If K is not symmetric, then Z K does not converge to K since the Z K are centrally symmetric by definition. However, all results can be extended to the non-symmetric case with minor modifications of the roofs by using the non-symmetric version of the L -centroid bodies from [] see also [6]. The aer is organized as follows. In section we give some bounds for the aroximation of volume in the case of a general convex body. In section 3 we consider the case of olytoes and we give the roof of Theorem.. Finally, in section 4, we discuss aroximation of a olytoe by -uniformly convex bodies see [] and we give the roof of Theorem..

3 Notation. We work in R n, which is equied with a Euclidean structure,. We denote by the corresonding Euclidean norm, and write B n for the Euclidean unit ball and S for the unit shere. Volume is denoted by. We write σ for the rotationally invariant surface measure on S. A convex body is a comact convex subset C of R n with non-emty interior. We say that C is symmetric, if x C imlies that x C. We say that C has center of mass at the origin if C x, θ dx = for every θ S. The suort function h C : R n R of C is defined by h C x = max{ x, y : y C}. C = {y R n : x, y for all x C} is the olar body of C. We refer to [] and [] for basic facts from the Brunn-Minkowski theory. Acknowledgments. The authors would like to thank the American Institute of Mathematics. Part of this work has been carried out during a stay at AIM. General Bounds Let K be a symmetric convex body in R n of volume. Let θ S. We define the arallel section function f K,θ : [ h k θ, h k θ] R + by f K,θ t := K θ + tθ. By Brunn s rincile, f K,θ is concave and attains its maximum at. So we have that t f K,θ f K,θt f K,θ. h K θ The right-hand side inequality is shar if and only if K is a cylinder in the direction of θ and the left-hand side inequality is shar if and only if K is a double cone in the direction of θ. The next roosition is well known. There, for x, y >, Bx, y = λx λ y dλ = ΓxΓy Γx+y function. is the Beta function and Γx = λ x e λ dλ is the Gamma Proosition.. Let K be a symmetric convex body in R n of volume. Let < and θ S. Then B +, n h ZKθ h K θ n. + Proof. As K =, n h Kθf K,θ h K θf K,θ. 3

4 Hence, on the one hand, with, hk h θ Z θ = K t f K,θ tdt f K,θ On the other hand, also with with, = hk θ + f K,θ h + n K θ + h K θ. hk h θ Z θ = K t f K,θ tdt f K,θ The roof is comlete. hk θ t dt t t dt h K θ = f K,θ h + K θ s s ds B +, nh K θ. As it was mentioned in the introduction, it was roved in [9] that if K is a C + symmetric convex body of volume, then log K K = nn + K. Before we consider the case of olytoes, we show that for every convex body we have that K K = O log. In articular, the following roosition holds. Proosition.. Let K be a symmetric convex body in R n of volume. Then n K log K K n K. Proof. We have that K K = n S h n Z θ K h n K θdσθ = h n K θ n S h n K θ h n Z θ dσθ, K where σ is the usual surface area measure on S. By Proosition., h n K θ n n n log = + ± o θ + log and h n Z K h n K θ h n Z θ B +, n n log n = + K ± o log. For the last equality see e.g. [9], Lemma which is also stated here as Lemma 3.3. Lebesgue s convergence theorem comletes the roof. 4

5 3 Polytoes Let K be a convex olytoe in R n with vertices v,..., v M. For k n, let A k = {F k : F k is a k-dimensional face of K}. For θ S and s h k θ let Let and Finally, for θ G K, let gθ, s = card {v i : v i K { v i, θ s}. B K = {θ S : s h K θ : gθ, s > } G K = {θ S : s < h K θ : gθ, s = } 3 s θ = min{s > : gθ, s = } 4 Remarks. Let θ G K. i Then there is a vertex v i such that for all s θ s h K θ {x K : x, θ s} = co [ K θ + sθ, v i ] ii Recall that f K,θ s = K θ + sθ. We have for all s θ s h K θ s h f K,θ s = f K,θ s θ K θ s 5 θ h K θ For a convex body K, let H K = max θ S h K θ. For k n, let K be a k-dimensional convex body in a k-dimensional affine sace of R n. Let be the inradius of K. Let rk = su{r > : x K such that x + rb k K} 6 r = min k min rf k F k A k Note that r >. We also ut h = max u BK h K u. For δ >, we define and Aδ = {θ S : u B K : θ u < δ}. 7 sδ = su θ S \Aδ s θ h K θ s θ h K θ Remark. sδ < and if θ φ where φ B K, then by continuity,. Hence we may assume that for δ > small enough, sδ is attained on the boundary of S \ Aδ. 8 5

6 Lemma 3.. Let K be a -symmetric olytoe in R n of volume. Then for δ small enough, s θ sδ = h K θ δr h su θ S \Aδ Proof. Let δ h H K. By the above Remark, for δ > small enough, there exists φ S \ Aδ such that sδ = s φ h K φ. As φ S \ Aδ, there exists u B K, such that u φ = δ. Let v K be that vertex of K such that φ, v = max x K φ, x. Let x = {αφ : α } K, z = {αu : α } K, and d = x z, d = x v. x, v and z lie in the -dimensional face F orthogonal to u. As φ G K, we may also assume that δ is small enough such that s φ = x, and hence sδ = x h K φ. Let ω be the angle between φ and u. Then tan ω = d h K u and sin ω = h Kφ s φ d. Hence and thus As d r and as δ Now observe that h K φ s φ d = d cos ω h K u s φ h K φ = d d cos ω h K uh K φ. d cos ω h K u, we get that s φ h K φ δ r h K φ. h k φ = h K φ u + h K u δh K + h K u h. Therefore, s φ h K φ δr h. Let f : R + R + be a C log-concave function with R + ftdt < and let. Let g t = t ft and let t = t f the unique oint such that g t =. We make use of the following Lemma due to B. Klartag [] Lemma 4.3 and Lemma

7 Lemma 3.. Let f be as above. For every ε,, t ftdt + Ce cε t +ε t ftdt t ε where C > and c > are universal constants. We will use Lemma 3. for the function f K,θ s = K θ + sθ in the roof of the next lemma. First we oberve Remark. Let θ G K. As above, let g t = t f K,θ t and let t be the unique oint such that g t =. Note that, since t h K θ, as see e.g. [9], Lemma 4.5, for large enough - namely so large that t s θ - we can use 5 and comute t. t = + n h Kθ 9 We will also use see e.g. [9], Lemma 4.3. Lemma 3.3. Let >. Then B +, n n = n log + n n4 log Γn + log n3 log Γn log ± o. Lemma 3.4. Let K be a -symmetric olytoe in R n of volume. For all sufficiently small δ, for all θ S \ Aδ and for all αnk δ, we have n hzkθ n log + n n log δ + c K,n h K θ. α n K = 4h r and c K,n are constants that deend on K and n only. Proof. Let < δ h H K be as in Lemma 3.. Let θ S \ Aδ. Hence, in articular, θ G K. By Lemma 3. we have for all ε, hk h θ Z θ = K t f K,θ tdt + Ce cε h K θ t f K,θ tdt εt Since t h K θ, as see e.g. [9], Lemma 4.5, there exists ε > which we will now determine, such that for all ε, εt s θ. 7

8 By 9, holds for all ε with By Lemma 3., sθ h K θ δr h We choose ε = rδ 4h. Then for ε n s θ h K θ ε s. θ h K θ and thus holds for all ε with ε n δ h δr r h ε/δ. ε n δ 4h r the estimate holds for all ε uniformly for all θ S \ Aδ. Thus, using also 5, h Z K θ + Ce cε h K θ t f K,θ tdt εt + Ce cε h K θ t f K,θ tdt s θ = + Ce cε h + K θf K,θs θ u u du s s θ θ h h K θ K θ + Ce cε h + K θf K,θ u u du s s θ θ h h K θ K θ n + Ce cε B +, n h K θ h. δr In the last inequality we have used that K =. Equivalently, becomes hzkθ h K θ With Lemma 3.3, we then get n n n + Ce cε n s θ h K θ δr h h δr n and that n h Kθf K,θ B +, n n. hzkθ h K θ n n log + n n log δ + c K,n. 8

9 Let δ [, and θ S. We define the ca Cθ, δ of the shere S around θ by Cθ, δ := {φ S : φ θ δ}. We will estimate the surface area of a ca, and to do so we will make use of the following fact which follows immediately from e.g. Lemma.3 in [3]. Lemma 3.5. Let θ S and δ <. Then vol B δ 4 δ σcθ, δ vol B δ 4 + δ4 4 δ. δ Proof of Theorem.. For given, let δ = log. Let Aδ as defined in.. Let be such that and δ = log satisfy the assumtions of Lemma 3.4, i.e. log 4h r. By Lemma 3.4, we have for all, K K n S \Aδ n log n S \Aδ h n Z θ K = n log n S h n Z θ K n log n h n Z θ K Aδ h n Z θ hn Z θ K K h n K θ dσθ log log n n + c K,n log log n n + c K,n log log n n + c K,n dσθ dσθ. dσθ Hence, log K K n h n Z θ K n S Aδ h n Z K θ n n n log log log + c K,n dσθ log n n n log log + c K,n log log dσθ. Note that, since K is centrally symmetric, rk = inf θ S h K θ. Also, since n Z K converges to K, for sufficiently large, h n Z θ rk K for every 9

10 θ S. Together with Lemma 3.5 we thus get n h n Z θdσθ K Aδ n+ n rk n card B K vol B δ n+ card B K vol n rk n B log. δ 4 + δ4 4 δ By Proosition. and Lebesgue s convergence theorem we can interchange integration and it and get n log K K S h n Z θ n n n log log K log n+ card B K vol B n rk n = n K. Here, we have also used that h ZKθ = h K θ. + c K,n dσθ log n n n log log log log n + c K,n log n The inequality from above follows by Proosition.. 4 Aroximation with uniformly convex bodies Let K be a symmetric convex body in R n and <. We say that K is -uniformly convex with constant C see e.g. [3, ], if for every x, y K, x + y K C x y K. We will need the following Proosition. The roof is based on Clarkson inequalities and can be found in e.g. [3],. 48. Proosition 4.. Let K be a comact set in R n of volume. Then for, Z K is -uniformly convex with constant C =. The symmetric difference metric between two convex bodies K and C is Proof of Theorem.. d s C, K = C \ K K \ C.

11 Let P = P P n. Then P = P n P and P = P P. Let K = P n Z P. Then by Proosition 4. we have that K is uniformly convex. Note that P K. By Theorem. we have that log P P = n P. So, for every ε >, there exists ε, P such that d s P, K = K P = P P P + εn P log P = + εn P log. We choose ε = and the roof is comlete. References [] R.J. Gardner, Geometric tomograhy, Encycloedia of Mathematics and its Alications, 58. Cambridge University Press, Cambridge, 995. [] R. J. Gardner and G. Zhang, Affine inequalities and radial mean bodies. Amer. J. Math. no.3 998, [3] D. J. H. Garling, Inequalities, A journey into Linear Analysis, Cambridge University Press 7. [4] E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. 78 no.3 999, [5] C. Haberl, Blaschke valuations, Amer. J. of Math., in ress [6] C. Haberl and F. Schuster, General L affine isoerimetric inequalities. J. Differential Geometry 83 9, -6. [7] C. Haberl, E. Lutwak, D. Yang and G. Zhang, The even Orlicz Minkowski roblem, Adv. Math. 4, [8] D. Klain, Star valuations and dual mixed volumes, Adv. Math. 996, 8-. [9] D. Klain, Invariant valuations on star-shaed sets, Adv. Math , [] B. Klartag, A central it theorem for convex sets, Invent. Math [] J. Lindenstrauss and L. Tzafriri, Classical Banach saces I, II, rerint, Sringer, 996.

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13 [9] E. Werner and D. Ye, Inequalities for mixed -affine surface area, Math. Ann. 347, [3] G. Zhang, New Affine Isoerimetric Inequalities, ICCM 7, Vol. II, Grigoris Paouris Deartment of Mathematics Texas A & M University College Station, TX,, U. S. A. Elisabeth Werner Deartment of Mathematics Université de Lille Case Western Reserve University UFR de Mathématique Cleveland, Ohio 446, U. S. A Villeneuve d Ascq, France elisabeth.werner@case.edu 3