Minkowski Problems for Geometric Measures

TL;DR

This survey unifies Minkowski type problems for a broad class of geometric measures arising in the Brunn-Minkowski theory and its dual, including surface area, cone-volume, dual curvature, and chord measures. It presents a cohesive variational and PDE-based framework, often reducing problems to Monge-Ampère type equations on the sphere and, in the dual setting, to flow approaches and entropy-based variational structures. The work covers classical results (Minkowski and Aleksandrov problems) and extensive developments in Lp, Orlicz, chord, dual, capacitary, and Gaussian settings, highlighting existence, uniqueness, and regularity, as well as self-similar Gauss curvature flows as a tool for regularity and construction. By tying together primal and dual theories, and connecting integral geometry with harmonic analysis, the paper reveals a unified landscape with rich interplays between variational methods, geometric inequalities, and geometric flows, while outlining key open problems and directions for future progress.

Abstract

This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and new Minkowski problems studied in convex geometry, PDEs, and harmonic analysis, and structured in a conceptual framework of the Brunn-Minkowski theory, its extensions, and related subjects.