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Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodies $L$. We prove the following sharp lower bound $$c \sqrt{n} \leq \mbox{lvr}(K),$$ for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{\frac{n}{\log \log(n)}}$. We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that $\mbox{lvr}(K)$ 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 $1 \leq p \leq \infty$), $K$ is the the unit ball of the full/symmetric tensor product of $\ell_p$-spaces endowed with the projective or injective norm or $K$ is unconditional.