Asymptotic estimates for the largest volume ratio of a convex body

Abstract

The largest volume ratio of a given convex body $K\subset {\mathbb{R}}^n$ is defined as

where the sup runs over all the convex bodies $L$. We prove the following sharp lower bound:

for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{n/\log \log (n)}$.

We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that $\mathrm{lvr}(K)$ 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 ${\mathbb{R}}^{d\times d}$ (e.g., the unit ball of the $p$-Schatten class $S_p^d$ for any $1\le p\le \infty$), if $K$ is the unit ball of the full/symmetric tensor product of ${\ell }_p$-spaces endowed with the projective or injective norm, or if $K$ is unconditional.