Acta Mathematica

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

Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang

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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.


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.

Article information

Acta Math., Volume 216, Number 2 (2016), 325-388.

Received: 10 June 2015
Revised: 10 June 2016
First available in Project Euclid: 30 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A38: Length, area, volume [See also 26B15, 28A75, 49Q20]
Secondary: 35J20: Variational methods for second-order elliptic equations

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

2016 © Institut Mittag-Leffler


Huang, Yong; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong. Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems. Acta Math. 216 (2016), no. 2, 325--388. doi:10.1007/s11511-016-0140-6.

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