$p$-Means of Convex Bodies: Sharpening Relations and Structural Properties

René Brandenberg; Florian Grundbacher

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arXiv · 2312.17512

$p$-Means of Convex Bodies: Sharpening Relations and Structural Properties

René Brandenberg·Florian Grundbacher

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Abstract

We study general $p$-means of convex bodies, extending the classical definitions by W. J. Firey via support and gauge functions to two families ranging over all $p \in [-\infty,\infty]$. For values of $p$ beyond the classical ranges, we show that $p$-means of polytopes are again polytopes, yielding simpler structural descriptions. Using a natural characterization of dilates of convex bodies based on their boundary structure, we characterize the equality cases between the two types of $p$-means for the same $p$-value. Extending recent results on standard mean-symmetrizations of convex bodies, we further establish (in almost all instances tight) inequalities quantifying how well arbitrary $p$-means of convex bodies approximate each other. These bounds lead to characterizations and sharp stability results for the equality cases between $p$-means for different $p$-values. As a corollary, every Minkowski centered convex body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance.

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René Brandenberg, Florian Grundbacher. 2026-06-16. $p$-Means of Convex Bodies: Sharpening Relations and Structural Properties. https://arxiv.org/abs/2312.17512

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