1. A Comprehensive Definition of the Geometric Mean of Convex Bodies Based on Relations Between Their $p$-Means
- Author
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Brandenberg, René and Grundbacher, Florian
- Subjects
Mathematics - Metric Geometry ,52A21 (Primary) 52A40 (Secondary) - 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., Comment: 31 pages, 4 figures
- Published
- 2023