Shadows of Convex Bodies

It is proved that if C is a convex body in R" then C has an affine image C (of nonzero volume) so that if P is any 1-codimensional orthogonal projection, \PC\>\C\{tt~l)/n. It is also shown that there is a pathological body, K , all of whose orthogonal projections have volume about \fh~ times as large as |Ä"| . 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 R" and 8 a unit vector, PeC will denote the orthogonal projection of C onto the 1-codimensional space perpendicular to 8 . Volumes and areas of convex bodies and their surfaces will be denoted with | • |. 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 «-dimensional Euclidean unit ball and let a = an_x be the rotationally invariant probability on the unit sphere S"~ . Cauchy's formula states that if C is a convex body in R" then its surface area is |ÔC| = ^ A x\PeC\da(8). The classical isoperimetric inequality in R" 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