Asymptotic estimates for the largest volume ratio of a convex body

Abstract

The largest volume ratio of given convex body K ⊂ R n is defined as lvr(K) := supwhere the sup runs over all the convex bodies L. We prove the following sharp lower bound c √ n ≤ lvr(K), for every body K (where c > 0 is an absolute constant). This result improves the former best known lower bound, of order n log log(n) . We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, if K is the unit ball of an unitary invariant norm in R d×d (e.g., the unit ball of the p-Schatten class S d p for any 1 ≤ p ≤ ∞) or K is unconditional, we show that lvr(K) behaves as the square root of the dimension of the ambient space.