Grunbaum's inequality for projections

摘要

Let K be a convex body in RTh whose centroid is at the origin, let E is an element of G(n,k) be a subspace, and let xi is an element of Sn-1 We find the best constant c = (k/n+1)(k) so that vertical bar(K vertical bar E) boolean AND xi(+)vertical bar(k) >= c vertical bar(K vertical bar E vertical bar(k) , and completely determine the minimizer. Here, 1k is k -dimensional volume, K vertical bar E is the projection of K onto E, and xi(+)= {x is an element of R-n : < x,xi > >= 0}. Our result generalizes both Griinbaurn's inequality, and an old inequality of Minkowsld and Radon. (C) 2016 Elsevier Inc. All rights reserved.