对偶 Brunn-Minkowski 理论中的几何测度及其关联的 Minkowski 问题
Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
AI总结：
本文回答了凸体 Federer 曲率测度的对偶类似物这一长期问题，并建立涉及测度集中性的充分条件，证明对偶 Minkowski 型问题解的存在性。
AI中文摘要：
对偶 Brunn-Minkowski 理论中一个长期存在的问题是：凸体的 Federer 曲率测度在对偶意义下的类似物是什么？本文给出了该问题的答案。这自然引出了 Minkowski 型问题的对偶版本，从而回答了如下问题：一个 Borel 测度成为凸体的对偶曲率测度的充分必要条件是什么？本文建立了涉及测度集中性的充分条件，以保证这些问题解的存在性。
英文摘要：
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.
