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 1≤k≤n and for each convex body K⊂Rn there is a centrally symmetric body L⊂Rn such that for any two projections P,Q:Rn→Rn of rank k one hasvr(PK,QL)≥cmin⁡{kn1log⁡log⁡log⁡(nlog⁡(n)k),klog⁡(nlog⁡(n)k)}, where c>0 is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime k≥n2/3.