An Improved Volume-Ratio Bound via Isotropic Positions

2026/08/03 by Daniel Galicer, Mariano Merzbacher, Damián Pinasco
Mathematics · #math.MG #math.FA #msc:52A23 #msc:52A38 #msc:52A40 #msc:46B06

paper · pdf

7 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

We show that, for every pair of convex bodies K,L⊂\mathbb Rⁿ, vr(K,L)≤ C√(nlog(n+1)). The main point is to place K and L^∘ 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 vr(K,L)≤ C√ n log(n+1) proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

Citations

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