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

• 成果类型: Article
• 署名作者: Huang, Yong; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong
• 署名单位: Hunan University; New York University; New York University Tandon School of Engineering
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-016-0140-6
• 发表日期: 2016
• 页码: 325-388
• 关键词: busemann-petty problem; intersection bodies; gauss curvature; surface measure; convex bodies; VALUATIONS; CLASSIFICATION; hypersurfaces; REGULARITY; sphere

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: 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.