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arXiv · 2608.02825

An Improved Volume Ratio Bound via Isotropic Positions

Daniel Galicer·Mariano Merzbacher·Damián Pinasco
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Abstract

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ 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 $ \operatorname{vr}(K,L)\leq 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.

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Daniel Galicer, Mariano Merzbacher, Damián Pinasco. 2026-08-03. An Improved Volume Ratio Bound via Isotropic Positions. https://arxiv.org/abs/2608.02825

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