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

Yong Huang Hunan university Erwin Lutwak New York University Deane Yang New York University Gaoyong Zhang New York University

Analysis of PDEs Differential Geometry Convex and Discrete Geometry mathscidoc:1702.03003

Silver Award Paper in 2017

Acta Mathematica, 216, (2), 325–388, 2016.11
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.
Monge–Ampere type equations, Minkowski problems, Alexandrov problem, curvature measures, area measures
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@inproceedings{yong2016geometric,
  title={Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems},
  author={Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170205230226945263183},
  booktitle={Acta Mathematica},
  volume={216},
  number={2},
  pages={325–388},
  year={2016},
}
Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang. Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems. 2016. Vol. 216. In Acta Mathematica. pp.325–388. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170205230226945263183.
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