  The difference body operator enjoys different characterization results
relying on its basic properties such as continuity, SL(n)-covariance, Minkowski
valuation or symmetric image. The Rogers-Shephard and the Brunn-Minkowski
inequalities provide upper and lower bounds for the volume of the difference
body in terms of the volume of the body itself. In this paper we aim to
understand the role of the Rogers-Shephard inequality in characterization
results of the difference body and, at the same time, to study the interplay
among the different properties. Among others, we prove that the difference body
operator is the only continuous and GL(n)-covariant operator from the space of
convex bodies to the origin-symmetric ones which satisfies a Rogers-Shephard
type inequality while every continuous and GL(n)-covariant operator satisfies a
Brunn-Minkowski type inequality.
