On the polar of Schneider's difference body

2025/03/08 by Haddad, Julián, Langharst, Dylan, Livshyts, Galyna V. +1 · 1 citation
#52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Secondary: 28A75

paper · doi:10.48550/arxiv.2503.06191

Abstract

In 1970, Schneider introduced the mth-order extension of the difference body DK of a convex body K⊂\mathbb Rⁿ, the convex body D^m(K) in \mathbb R^nm. He conjectured that its volume is minimized for ellipsoids when the volume of K is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of D^m(K) is maximized precisely by ellipsoids. For m=1 this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version.

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