The role of the Rogers-Shephard inequality in the characterization of the difference body (Q1683658)

A general classification result for operators on convex bodies of type \(aK+b(-K)\), where \(K\) is a convex body in \(\mathbb{R}^n\) and \(a, b\geq 0\), is provided, having as important consequence the fact that the difference body operator is the only continuous and \(\mathrm{GL}(n)\)-covariant operator from the space of convex bodies to the origin-symmetric ones that satisfies a Rogers-Shephard type inequality. Several illustrative examples and hints towards a forthcoming work in the same area close the paper.