Asymptotic estimates for the largest volume ratio of a convex body

Abstract

The largest volume ratio of a given convex body K ⊂ Rn is defined as lvr(K) := sup L⊂Rn vr(K, L), where 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 behaviour of the largest volume ratio for some natural classes. In particular, we show that lvr(K) behaves as the square root of the dimension of the ambient space in the following cases: if K is the unit ball of an unitary invariant norm in Rd×d (e.g., the unit ball of the p-Schatten class Sd p for any 1 ≤ p ≤ ∞), if K is the unit ball of the full/symmetric tensor product of p-spaces endowed with the projective or injective norm, or if K is unconditional.