The role of the Rogers–Shephard inequality in the characterization of the difference body

摘要

The Rogers–Shephard and 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 work it is shown that the difference body operator is the only continuous and GL ? ( n ) {operatorname{GL}(n)} -covariant operator from the space of convex bodies to the origin-symmetric ones which satisfies a Rogers–Shephard-type inequality.This is a consequence of a more general classification result for operators on convex bodies of the type a ? K + b ? ( - K ) {aK+b(-K)} .