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

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Acta Mathematica

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.

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Authors and Affiliations

  1. Institute of Mathematics, Hunan University, Changsha, 410082, China

    Yong Huang

  2. Department of Mathematics, New York University, Tandon School of Engineering, 6 Metrotech Center, Brooklyn, NY, 11201, U.S.A.

    Erwin Lutwak, Deane Yang & Gaoyong Zhang

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  1. Yong Huang
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  2. Erwin Lutwak
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  3. Deane Yang
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  4. Gaoyong Zhang
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Correspondence to Yong Huang.

Additional information

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.

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Huang, Y., Lutwak, E., Yang, D. et al. Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems. Acta Math 216, 325–388 (2016). https://doi.org/10.1007/s11511-016-0140-6

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  • DOI: https://doi.org/10.1007/s11511-016-0140-6

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