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: 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.

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对偶 Brunn-Minkowski 理论中的几何测度及其相关的 Minkowski 问题

对偶 Brunn-Minkowski 理论中一个长期存在的问题是“费德勒的凸体曲率测量的对偶类似物是什么？” 对此提供了答案。这自然会导致 Minkowski 型问题的对偶版本：Borel 测度成为凸体的对偶曲率测度的充分必要条件是什么？为这些问题的解决方案的存在建立了足够的条件，包括测量浓度。