Intersection bodies and dual mixed volumes

Abstract

The setting for this article is Euclidean n-space, KY’, n > 2. Let X” denote the set of convex bodies (compact, convex subsets with nonempty interiors) in R”, and let -X; denote the subset of X” that contains the centered (centrally symmetric with respect to the origin) bodies. For UE S”‘, let E, denote the hyperplane, through the origin, that is orthogonal to U, and let KI E, denote the image of the orthogonal projection of the body KE X” onto E,. We shall use V, to denote i-dimensional Lebesgue measure. For V,, and V,-, we shall usually write V and u. Shephard [38] poses the question: If K, L E XE, and