Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 2

The logarithmic Minkowski problem in $R^2$

Pages: 869 – 902

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n2.a5

Authors

Yude Liu (School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, China)

Xinbao Lu (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

Qiang Sun (School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, China)

Ge Xiong (School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, China)

Abstract

A necessary condition for the existence of solutions to the logarithmic Minkowski problem in $\mathbb{R}^2$, which turns to be stronger than the celebrated subspace concentration condition, is given. The sufficient and necessary conditions for the existence of solutions to the logarithmic problem for quadrilaterals, as well as the number of solutions, are fully characterized.

Keywords

logarithmic Minkowski problem, cone-volume measure, subspace concentration condition, polytope

2010 Mathematics Subject Classification

35J60, 52A40

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Research of the second author is supported by the China Postdoctoral Science Foundation No. 2023M730698, and by the China National Funded Postdoctoral Researcher Program No. GZB20230165.

Research of the fourth author is supported by NSFC No. 12271407.

Received 4 June 2023

Received revised 17 August 2023

Accepted 11 September 2023

Published 3 April 2024