  Sumset estimates, which provide bounds on the cardinality of sumsets of
finite sets in a group, form an essential part of the toolkit of additive
combinatorics. In recent years, probabilistic or entropic analogs of many of
these inequalities were introduced. We study analogues of these sumset
estimates in the context of convex geometry and Lebesgue measure on ${\mathbb
R}^n$. First, we observe that, with respect to Minkowski summation, volume is
supermodular to arbitrary order on the space of convex bodies. Second, we
explore sharp constants in the convex geometry analogues of variants of the
Pl\"unnecke-Ruzsa inequalities. In the last section of the paper, we provide
connections of these inequalities to the classical Rogers-Shephard inequality.
