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.
From: Florian Grundbacher [view email]
[v1]
Fri, 29 Dec 2023 08:19:00 UTC (35 KB)
[v2]
Wed, 1 May 2024 14:14:41 UTC (38 KB)
[v3]
Tue, 16 Jun 2026 09:57:59 UTC (38 KB)
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