Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of a given convex body $K \subset \mathbb R^n$ is defined as $$ \mathrm{lvr}(K):= \mathrm {sup}\_{L R^n} {vr}(K,L), where the sup runs over all bodies $L$. We prove following sharp lower bound: c \sqrt{n} \leq \mathrm{lvr}(K), for every $K$ (where $c > 0$ an absolute constant). This result improves former best known bound, order $\sqrt{{n}/{\log \log(n)}}$. also study exact asymptotic behaviour some natural classes. In particular, we show that $\mathrm{lvr}(K)$ behaves square root dimension ambient space in cases: if unit ball unitary invariant norm $\mathbb{R}^{d \times d}$ (e.g., $p$-Schatten class $S\_p^d$ any $1 p \infty$), full/symmetric tensor product $\ell\_p$-spaces endowed with projective or injective norm, unconditional.