Kubota-type formulas and supports of mixed measures

Abstract

Motivated by a problem for mixed Monge–Ampere measures of convex func- ` tions, we address a special case of a conjecture of Schneider and show that for every convex body K the support of the mixed area measure S(K[ j],B n−1 L [n− 1− j],·) is given by the set of (K[ j],B n−1 L [n− 1− j])-extreme unit normal vectors, where B n−1 L denotes the (n− 1)-dimensional Euclidean unit ball in a hyperplane L. As a consequence, we see that the supports of these measures are nested. Our proof introduces a Kubota-type formula for mixed area measures, which involves in tegration over j-dimensional linear subspaces that contain a fixed 1-dimensional subspace. We transfer these results to the analytic setting, where we obtain cor responding statements for (conjugate) mixed Monge–Ampere measures of convex ` functions. Thereby, we establish a fundamental property for functional intrinsic volumes. In addition, we study connections between mixed Monge–Ampere mea- ` sures of convex functions and mixed area measures and mixed volumes of convex bodies.