  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.
