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

TL;DR

This paper identifies dual curvature measures in the dual Brunn-Minkowski theory, establishing conditions for their existence and formulating dual Minkowski problems, thus advancing the understanding of convex geometric measures.

Contribution

It provides the first characterization of dual curvature measures and formulates dual Minkowski problems with existence conditions, extending classical convex geometry theories.

Findings

Established dual curvature measures for convex bodies.

Derived necessary and sufficient conditions for dual Minkowski problems.

Proved existence of solutions under measure concentration conditions.

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.