Acta Mathematica

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

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

Full-text: Open access

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.

Note

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802481

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

Zentralblatt MATH identifier
06668372

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

Keywords
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

Rights
2016 © Institut Mittag-Leffler

Citation

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. https://projecteuclid.org/euclid.acta/1485802481.


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