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In light of the log-Brunn-Minkowski conjecture, various attempts have beenmade to define the geometric mean of convex bodies. Many of these constructionsare fairly complex and/or fail to satisfy some natural properties one wouldexpect of such a mean. We remedy this by providing a new geometric mean that isboth technically simple and inherits all the natural properties expected. Toimprove our understanding of potential geometric mean definitions, we firststudy general $p$-means of convex bodies, with the usual definition extended totwo series ranging over all $p \in [-\infty,\infty]$. We characterize theirequality cases and obtain (in almost all instances tight) inequalities thatquantify how well these means approximate each other. As a corollary, weestablish that every Minkowski centered body is equidistant from all its$p$-symmetrizations with respect to the Banach-Mazur distance. Finally, we showthat our geometric mean satisfies all the properties considered in recentliterature and extend this list with some properties regarding symmetrizationand asymmetry.

Relations Between $p$-Means of Convex Bodies and a New Suggestion for the Definition of the Geometric Mean