An Improved Volume Ratio Bound via Isotropic Positions

We show that, for every pair of convex bodies $K,L\subset {\mathbb{R}}^n$, $$\mathrm{vr}(K,L)\le C\sqrt{n\log (n+1)}\mathrm{.}$$ The main point is to place $K$ and $L^{\circ }$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin’s recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $\mathrm{vr}(K,L)\le C\sqrt{n}\log (n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.