p-Cross-section Bodies

If K is a convex body in En, its cross-section body CK has a radial function in any direction u is ∈ Sn-1 equal to the maximal volume of hyperplane sections of K orthogonal to u. A generalization called the p-cross-section body CpK of K, where p > -1, is introduced. The radial function of CpK in any direction u ∈ Sn-1 is the pth mean of the volumes of hyperplane sections of K orthogonal to u through points in K. It is shown that C1K is convex but CpK is generally not convex when p > 1. An inclusion of the form an,qCqK ⊆ an,pCpK, where -1 < p < q and the constant an,p is the best possible, is established. This is applied to disprove a conjecture of Makai and Martini.