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

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

Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	325-388
Number of pages	64
Journal	Acta Mathematica
Volume	216
Issue number	2
DOIs	https://doi.org/10.1007/s11511-016-0140-6
State	Published - Jun 1 2016

Keywords

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

ASJC Scopus subject areas

General Mathematics

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10.1007/s11511-016-0140-6

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abstract = "A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer{\textquoteright}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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AU - Lutwak, Erwin

AU - Yang, Deane

AU - Zhang, Gaoyong

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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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KW - Alexandrov problem

KW - L-Minkowski problem

KW - Minkowski problem

KW - cone-volume measure

KW - dual Brunn–Minkowski theory

KW - dual curvature measure

KW - integral curvature

KW - logarithmic Minkowski problem

KW - surface area measure

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

Abstract

Keywords

ASJC Scopus subject areas

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