Estimating the Sizes of Convex Bodies from Projections

Abstract

Let 1 ⩽ r ⩽ s ⩽ d −1, and let Ls = (L1, α1;…; Ln, αn) be a finite family of s-dimensional linear subspaces Li of Ed, with associated positive weights αi, for i = 1,…,n. Denote by Vr(K,L) the intrinsic r-volume of the image of the at least r-dimensional compact convex set K under orthogonal projection on to L, write and let Vr(K) be the intrinsic r-volume of K. (The intrinsic r-volumes, which are normalized quermassintegrals, are measures of the sizes of convex sets in various senses.) In this paper is considered the problem of determining bounds for the ratio πr(K, Ls) = Vr(K, Ls)/Vr(K), or, in other words, the stereometric problem of how good an estimator Vr(K, Ls) is for Vr(K). In the case when r = s = 1 and L1 has a large amount of symmetry, a complete solution is given. In the case when r = s = d − 1; there is also a solution which is, in principle, complete. More generally, estimates can be made when s = d − 1; this problem is related to another on approximating the unit ball by zonotopes.