Mathematics > Metric Geometry
[Submitted on 29 Dec 2023 (v1), last revised 1 May 2024 (this version, v2)]
Title:A Comprehensive Definition of the Geometric Mean of Convex Bodies Based on Relations Between Their $p$-Means
View PDF HTML (experimental)Abstract:In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of such a mean. We remedy this by providing a new geometric mean that is both technically simple and inherits all the natural properties expected. To improve our understanding of potential geometric mean definitions, we first study general $p$-means of convex bodies, with the usual definition extended to two series ranging over all $p$ in the extended reals. We characterize their equality cases and obtain (in almost all instances tight) inequalities that quantify how well these means approximate each other. As a corollary, we establish that every Minkowski centered body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance. Finally, we show that our geometric mean satisfies all the properties considered in recent literature and extend this list with some properties regarding symmetrization and asymmetry.
Submission history
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)
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