Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang
TL;DR
The paper develops the dual Brunn-Minkowski theory by defining dual curvature measures $\overset{\sim}{C}_q(K,\cdot)$ as the differential objects of dual quermassintegrals $\overset{\sim}{W}_{n-q}(K)$ and formulates the dual Minkowski problem in this unified framework. It proves a dual Aleksandrov-type variational formula for the entire range of dual quermassintegrals, and reduces the dual Minkowski problem to a variational maximization problem for a functional $\Phi_\mu$, establishing existence of solutions in the origin-symmetric setting under a $q$-subspace mass inequality. The work unifies the logarithmic Minkowski and Aleksandrov problems as special cases of the dual problem and provides new tools, including entropy-type inequalities and non-degeneracy criteria, to ensure existence. This advances the understanding of dual geometric measures, their PDEs, and their associated extremal problems, with potential implications for affine/invariant inequalities and dual PDE approaches in convex geometry.
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.