On the volume ratio of projections of convex bodies

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 subset of R-n there is a centrally symmetric body L subset of R-n such that for any two projections P, Q : R-n - R-n of rank k one has vr(PK, QL) >= c min {k/root n ,V root 1/log log log(n log(n)/k ), root k/root log(n log(n) k)}, where c > 0 is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime k >= n(2/3). (c) 2023 Elsevier Inc. All rights reserved.