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

Yong Huang; Erwin Lutwak; Deane Yang; Gaoyong Zhang

doi:10.1007/s11511-016-0140-6

2016 Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang

Author Affiliations +

Acta Math. 216(2): 325-388 (2016). DOI: 10.1007/s11511-016-0140-6

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: 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.

Funding Statement

Research of the first author supported, in part, by NSFC No.11371360; research of the other authors supported, in part, by NSF Grant DMS-1312181.

Citation

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Yong Huang. Erwin Lutwak. Deane Yang. Gaoyong Zhang. "Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems." Acta Math. 216 (2) 325 - 388, 2016. https://doi.org/10.1007/s11511-016-0140-6

Information

Received: 10 June 2015; Revised: 10 June 2016; Published: 2016

First available in Project Euclid: 30 January 2017

zbMATH: 1372.52007

MathSciNet: MR3573332

Digital Object Identifier: 10.1007/s11511-016-0140-6

Subjects:

Primary: 52A38

Secondary: 35J20

Keywords: $L_p$-Minkowski problem , Alexandrov problem , cone-volume measure , dual Brunn–Minkowski theory , dual curvature measure , integral curvature , logarithmic Minkowski problem , Minkowski problem , surface area measure

Rights: 2016 © Institut Mittag-Leffler

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