Affine inequalities and radial mean bodies

We introduce for p > - 1 the radial pth mean body RpK of a convex body K in double-struckⁿ. The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to p = ∞ and p = -1, respectively. We prove that RpK is convex when p > 0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RqK when -1 < p < q. When p = n and q → ∞, this becomes the Rogers-Shephard inequality, and when p → -1 and q = n, it becomes a reverse form of the Petty projection inequality proved previously by the second author.