Shadows of convex bodies
Shadows of convex bodies
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DOI：
10.1090/s0002-9947-1991-1035998-3
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发表时间：
1989-10
影响因子：
1.3
通讯作者：
K. Ball
中科院分区：
文献类型：
--
作者：
K. Ball
证明了：若C是Rn中的凸体，则C有仿射像C（体积非零），使得若P是任意1-余维正交投影，|PCa > 1 × c1（n-1）/n还表明存在一个病态体K，其正交投影的体积约为VX/x ×伊基（n-1）/n 0。引言本文讨论的问题涉及的领域的阴影（正交投影）的凸体，并在较小程度上，这种机构的表面积。如果C是iRn中的凸体，0是单位向量，则P. C将表示C在1-余维空间上垂直于0的正交投影。凸体及其表面的面积和面积将用I1表示。凸体的阴影和表面积之间的关系用柯西的著名公式表示。对于每个n E N，设vn是n维欧几里得单位球的体积，并且设a = an-1是单位球Sn上的旋转不变概率。柯西公式指出，如果C是Rn中的凸体，则其表面积为=ac I S 'P6 CI dcr（O）。经典的Rn等周不等式指出，任何物体的表面积至少与相同体积的欧几里得球一样大。本文的第一部分致力于证明一个“局部”等周不等式，证明所有物体都有大的阴影，而不仅仅是大的表面积（或平均阴影）。这个结果的主要动机是它与Vaaler猜想的关系以及围绕它的重要问题。这个定理及其与Vaaler猜想的关系在§ 1的开头描述。所谓凸体的“投影体”在凸体理论中起着重要的作用。这是很容易看到的，通过考虑多面体，1989年9月12日由编辑收到。1980年数学学科分类（1985年修订）。初级52 A20，52 A40。作者得到了NSF DMS-8807243的部分支持。（1991年美国数学学会0002-9947/91）每页1.00美元+0.25美元
It is proved that if C is a convex body in Rn then C has an affine image C (of nonzero volume) so that if P is any 1-codimensional orthogonal projection, |PCa > 1Žcl(n-l)/n It is also shown that there is a pathological body, K, all of whose orthogonal projections have volume about VX/ times as large as IKI(n 1)/n 0. INTRODUCTION The problems discussed in this paper concern the areas of shadows (orthogonal projections) of convex bodies and, to a lesser extent, the surface areas of such bodies. If C is a convex body in iRn and 0 a unit vector, P. C will denote the orthogonal projection of C onto the 1-codimensional space perpendicular to 0. Volumes and areas of convex bodies and their surfaces will be denoted with I 1. The relationship between shadows and surface areas of convex bodies is expressed in Cauchy's well-known formula. For each n E N, let vn be the volume of the n-dimensional Euclidean unit ball and let a = an-, be the rotationally invariant probability on the unit sphere Sn. Cauchy's formula states that if C is a convex body in Rn then its surface area is =ac I S'P6CI dcr(O). vnI Jn I The classical isoperimetric inequality in Rn states that any body has surface area at least as large as a Euclidean ball of the same volume. The first section of this paper is devoted to the proof of a "local" isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area (or average shadow). The principal motivation for this result is its relationship to a conjecture of Vaaler and the important problems surrounding it. This theorem and its connection with Vaaler's conjecture are described at the beginning of § 1. An important role is played in the theory of convex bodies by the so-called "projection body" of a convex body. It is easily seen, by considering polytopes, Received by the editors September 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 52A20, 52A40. The author was supported in part by NSF DMS-8807243. ( 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page