Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems

Abstract

A longstanding question in the dual Brunn-Minkowski theory is what are the dual analogues of Federer's curvature measures for convex bodies. The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems, which answer the question of what are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body. Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.