On inner quermasses of convex bodies (Q1098419)

Let K be a convex body in \({\mathbb{R}}^ d\) (d\(\geq 3)\) and \b{V}\({}_{d- 1}(K,u)\) denote the inner (d-1)-quermass of K in direction u, i.e. the maximum area of the intersection of K and some hyperplane orthogonal to u. It is shown that \b{V}\({}_{d-1}(K,u)\) is smaller than or equal to the corresponding reciprocal value of the distance function of the projection body \(\Pi\) K of K. Moreover, the d-ellipsoids are characterized by equality for each direction. These relations imply elementary statements on the intersection body IK of K, which was recently introduced by E. Lutwak. Additionally, some unsolved problems concerning quermasses of convex bodies are formulated.