Lp dual curvature measures

Abstract

A new family of geometric Borel measures on the unit sphere is introduced. Special cases include the Lp surface area measures (which extend the classical surface area measure of Aleksandrov and Fenchel & Jessen) and Lp-integral curvature (which extends Alkesandrov's integral curvature) in the Lp Brunn–Minkowski theory. It also includes the dual curvature measures (which are the duals of Federer's curvature measures) in the dual Brunn–Minkowski theory. This partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes.