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The discrete Orlicz chord Minkowski problem

Part of: General convexity

Published online by Cambridge University Press:  09 December 2024

Suwei Li  and
Hailin Jin
Suwei Li
Affiliation:
Department of Mathematics, Suzhou University of Science and Technology, Suzhou, 215009 China e-mail: lisuwei152@163.com
Hailin Jin*
Affiliation:
Department of Mathematics, Suzhou University of Science and Technology, Suzhou, 215009 China e-mail: lisuwei152@163.com
*
e-mail: jinhailin17@163.com
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Abstract

In this paper, we consider the discrete Orlicz chord Minkowski problem and solve the existence of this problem, which is the nontrivial extension of the discrete $L_{p}$ chord Minkowski problem for ${0<p<1}$.

Keywords

Chord integralchord measurechord Minkowski problem

MSC classification

Primary: 52A40: Inequalities and extremum problems

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 154 - 165
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Project supported by National Nature Science Foundation of China No. 12071334 and No. 12071277.

References

Bianchi, G., Böröczky, K. J., Colesanti, A. and Yang, D., The planar ${L}_p$ -Minkowski problem for $-n<p<1$ . Adv. Math. 341(2019), 493–535.CrossRefGoogle Scholar
Böröczky, K. J., Lutwak, E., Yang, D. and Zhang, G., The logarithmic Minkowski problem . J. Amer. Math. Soc. 26(2013), 831–852.CrossRefGoogle Scholar
Böröczky, K. J. and Trinh, H. T., The planar ${L}_p$ -Minkowski problem for $0<p<1$ . Adv. in Appl. Math. 81(2017), 58–81.CrossRefGoogle Scholar
Chen, S., Li, Q. and Zhu, G., On the ${L}_p$ Monge-Ampé re equation. J. Differential Equations. 263(2017), 4997–5011.CrossRefGoogle Scholar
Chou, K.-S. and Wang, X.-J., The ${L}_p$ -Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205(2006), 33–83.CrossRefGoogle Scholar
Gardner, R. J., Hu, D. and Weil, W., The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities . J. Differential Geom. 97(2014), 427–476.CrossRefGoogle Scholar
Gardner, R. J., Hug, D., Weil, W., Xing, S. and Ye, D., General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I . Calc. Var. Partial Differential Equations. 58(2019), 35 pp.CrossRefGoogle Scholar
Gardner, R. J., Hug, D., Xing, S. and Ye, D., General volumes in the Orlica-Brunn-Minkowski theory and a related Minkowski problem II . Calc. Var. Partial Differential Equations 59 (2020), 33 pp.CrossRefGoogle Scholar
Guan, P. and Xia, C., The ${L}_p$ Christoffel-Minkowski problem: the case $1<p<k+1$ . Calc. Var. Partial Differential Equations 57(2018), 23 pp.CrossRefGoogle Scholar
Guo, L., Xi, D. and Zhao, Y., The ${L}_p$ chord Minkowski problem in a critical interval. Math. Ann. (2023), https://doi.org/10.1007/s00208-023-02721-8.Google Scholar
Haberl, C., Lutwak, E., Yang, D. and Zhang, G., The even Orlicz Minkowski problem . Adv. Math. 224(2010), 2485–2510.CrossRefGoogle Scholar
Hu, J. R., Huang, Y. and Lu, J., On the regularity of the chord log-Minkowski problem. Preprint, 2023, arXiv:2304.14220.Google Scholar
Hu, J. R., Huang, Y., Lu, J. and Wang, S. N., The chord gauss curvature flow and its ${L}_p$ chord Minkowski problem. Preprint, 2023, arXiv:2305.00453.Google Scholar
Huang, Q. and He, B., On the Orlicz Minkowski problem for polytopes . Discrete Comput. Geom. 48(2012), 281–297.CrossRefGoogle Scholar
Huang, Y., Lutwak, E., Yang, D. and Zhang, G., The ${L}_p$ Alesandrov problem for ${L}_p$ integral curvature. J. Differential Geom. 110(2018), 1–29.CrossRefGoogle Scholar
Huang, Y. and Zhao, Y., On the ${L}_p$ dual Minkowski problem. Adv. Math. 332 (2018), 57–84.CrossRefGoogle Scholar
Hug, D., Lutwak, E., Yang, D. and Zhang, G., On the ${L}_p$ Minkowski problem for polytopes. Discrete Comput. Geom. 33(2005), 699–715.CrossRefGoogle Scholar
Li, Y., The ${L}_p$ chord Minkowski problem for negative $p$ . J. Geom. Anal. 34(2024), Paper No. 82, 23.CrossRefGoogle Scholar
Lutwak, E., The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem . J. Differential Geom. 38(1993), 131–150.Google Scholar
Lutwak, E., Xi, D., Yang, D. and Zhang, G., Chord measures in integral geometry and their Minkowski problems . Comm. Pure Appl. Math. (2023), https://doi.org/10.1002/cpa.22190.Google Scholar
Lutwak, E., Yang, D., and Zhang, G., Orlicz projection bodies. Adv. Math. 223(2010), 220–242.CrossRefGoogle Scholar
Wu, Y., Xi, D. and Leng, G., On the discrete Orlicz Minkowski problem . Trans. Amer. Math. Soc. 371(2019), 1795–1814.CrossRefGoogle Scholar
Xi, D., Jin, H. and Leng, G., The Orlicz Brunn-Minkowski inequality . Adv. Math. 260(2014) 350–374.CrossRefGoogle Scholar
Xi, D., Yang, D., Zhang, G. and Zhao, Y., The ${L}_p$ chord Minkowski problem. Adv. Nonlinear Stud. 23 (2023), 22 pp.CrossRefGoogle Scholar
Xie, F., The Orlicz Minkowski problem for general measures . Proc. Amer. Math. Soc. 150(2022), 4433–4445. Google Scholar
Xie, F., The Orlicz Minkowski problem for cone-volume measure . Adv. in Appl. Math. 149(2023), Paper No. 102523, 18.Google Scholar
Zhao, X. and Zhao, P., Flow by Gauss curvature to the Orlicz chord Minkowski problem. Preprint, 2023, arXiv.2308.05332v2.CrossRefGoogle Scholar