# Asymptotic estimates for the largest volume ratio of a convex body

### Daniel Galicer

Universidad de Buenos Aires, Argentina### Mariano Merzbacher

Universidad de Buenos Aires, Argentina### Damián Pinasco

Universidad T. Di Tella, Buenos Aires, Argentina

## 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 $1 \leq p \leq \infty$), 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.

## Cite this article

Daniel Galicer, Mariano Merzbacher, Damián Pinasco, Asymptotic estimates for the largest volume ratio of a convex body. Rev. Mat. Iberoam. 37 (2021), no. 6, pp. 2347–2372

DOI 10.4171/RMI/1263