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Transactions of the American Mathematical Society

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The logarithmic Minkowski problem for non-symmetric measures


Authors: Shibing Chen, Qi-rui Li and Guangxian Zhu
Journal: Trans. Amer. Math. Soc. 371 (2019), 2623-2641
MSC (2010): Primary 52A40, 35J96, 35J75
DOI: https://doi.org/10.1090/tran/7499
Published electronically: November 16, 2018
MathSciNet review: 3896091
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Abstract: Recently, Böröczky, Lutwak, Yang, and Zhang [J. Amer. Math. Soc. 26 (2013), pp. 831-852] established necessary and sufficient conditions for the existence of solutions to the logarithmic Minkowski problem for origin-symmetric measures. We establish the existence of solutions to the logarithmic Minkowski problem for non-symmetric measures and demonstrate the non-uniqueness of solutions to the logarithmic Minkowski problem.


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Additional Information

Shibing Chen
Affiliation: School of Mathematical Sciences, University of Science and Technology of China, No. 96, JinZhai Road Baohe District, Hefei, Anhui, 230026, People’s Republic of China
Email: chenshib@ustc.edu.cn

Qi-rui Li
Affiliation: Mathematical Science Institute, The Australian National University, Canberra ACT 2601
Email: Qi-Rui.Li@anu.edu.au

Guangxian Zhu
Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
Email: gzhu@cims.nyu.edu

DOI: https://doi.org/10.1090/tran/7499
Keywords: Cone-volume measure, logarithmic Minkowski problem, subspace concentration condition
Received by editor(s): November 26, 2016
Received by editor(s) in revised form: May 23, 2017, and September 24, 2017
Published electronically: November 16, 2018
Additional Notes: The corresponding author: Guangxian Zhu.
Article copyright: © Copyright 2018 American Mathematical Society