On the volume ratio of projections of convex bodies

Abstract

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body K we show that there is another convex body L such that the volume ratio between any two projections of fixed rank of the bodies K and L is large. Namely, we prove that for every and for each convex body there is a centrally symmetric body such that for any two projections of rank k one has where is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime .