Asia-Pacific Journal of Mathematics and Statistics

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The Classical Minkowski Problem: An Optimization Description (A Brief Review)

Received: 10 December 2019    Accepted: 03 March 2020    Published: 16 March 2020
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Abstract

The research objective of this brief review paper is trying to introduce an optimization description for the classical (discrete) Minkowski problem to our mathematical readers, in the use of as plain and simple mathematical languages as possible. We also summarized some breakthroughs of the research history of classical Minkowski problem, and look forward to seeing a current breakthrough proposed. The classical Minkowski problem can be reduced to find solutions of the famous Monge-Ampere equation. The optimal transportation problem can be reduced to find solutions of the famous Monge-Ampere equation too. Optimal transport theory is an important application of the Minkowski problem. This brief review paper introduced the optimal transportation problem and its applications at the end of this paper. At present, the Monge–Kantorovich optimal transport has found applications in wide range in different fields (including image registration and warping, reflector design, retrieving information from shadowgraphy and proton radiography, seismic tomography and reflection seismology, etc.). In conclusion, the author of this brief review paper described and studied the classical Minkowski problem in optimization languages. Minkowski duality is also needed to study furthermore. This is an interesting research topic for optimization related researchers. There is a wealth of information and insight in the classical Minkowski problem and its optimal transportation problem. This studying is worth the effort.

Published in Asia-Pacific Journal of Mathematics and Statistics (Volume 2, Issue 1, March 2020)
Page(s) 1-4
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Classical Minkowski Problem, Discrete Minkowski Problem, Convex Analysis, Optimization Theory, History and Open Questions

References
[1] Schneider R (1993) Convex bodies: the Brunn-Minkowski theory (encyclopedia of mathematics and its applications), Cambridge: Cambridge University Press.
[2] Gu XF (2016) Optimal Transport Theory Lecture Note 3 (in Chinese): http://blog.sciencenet.cn/blog-2472277-954180.html. Accessed on 25/02/2016.
[3] Gu X, Luo F, Sun J, Yau S-T (2013) Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge-Ampere equations: https://arxiv.org/pdf/1302.5472.pdf. Accessed on 29/11/2019.
[4] Su ZY, Sun J, Gu XF, Luo F, Yau S-T (2014) Optimal mass transport for geometric modeling based on variational principles in convex geometry. Engineering with Computers 30 (4) 475–486.
[5] Merigot Q (2011) A multiscale approach to optimal transport. Computer Graphics Forum 30 (5) 1584–1592.
[6] https://en.wikipedia.org/wiki/Minkowski_problem, https://en.wikipedia.org/wiki/Talk:Minkowski_problem, Accessed on 29/11/2019.
[7] Nirenberg L (1953) The Weyl and Minkowski problems in differential geometry in the large. Communications on Pure and Applied Mathematics 6 (3) 337-394.
[8] Busemann H (1959) Minkowski’s and related problems for convex surfaces with boundaries. Michigan Mathematical Journal 6 (3) 259-66.
[9] Pogorelov AV (1979) The Minkowsky Multidimensional Problem, Washington: Scripta, ISBN 0470–99358–8.
[10] Cheng S-Y, Yau S-T (1976) On the regularity of the solution of the n-dimensional Minkowski problem. Communications on Pure and Applied Mathematics 29 (5) 495-516.
[11] Bodrenko AI (2007) The solution of the Minkowski problem for open surfaces in Riemannian space: https://arxiv.org/pdf/0708.3929.pdf. Accessed on 29/11/2019.
[12] Monge G (1781) Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année (Mem Math Phys Acad Royale Sci) 666–704.
[13] Kantorovich L (1942) On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37: 199–201.
[14] Villani C (2003) Topics in Optimal Transportation. American Mathematical Soc. p. 66. ISBN 978-0-8218-3312-4.
[15] Rao SS (2009) Engineering Optimization: Theory and Practice (4th ed.). John Wiley & Sons. p. 221. ISBN 978-0-470-18352-6.
[16] Kutateladze SS, Rubinov AM (1972) Minkowski duality and its applications. Russian Mathematical Surveys 27 (3) 137-191.
Author Information
  • Centre of Informatics and Applied Optimisation, the Federation University Australia, Ballarat, Australia

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  • @article{10045508,
      author = {Jiapu Zhang},
      title = {The Classical Minkowski Problem: An Optimization Description (A Brief Review)},
      journal = {Asia-Pacific Journal of Mathematics and Statistics},
      volume = {2},
      number = {1},
      pages = {1-4},
      url = {http://www.sciencepg.com/article/10045508},
      abstract = {The research objective of this brief review paper is trying to introduce an optimization description for the classical (discrete) Minkowski problem to our mathematical readers, in the use of as plain and simple mathematical languages as possible. We also summarized some breakthroughs of the research history of classical Minkowski problem, and look forward to seeing a current breakthrough proposed. The classical Minkowski problem can be reduced to find solutions of the famous Monge-Ampere equation. The optimal transportation problem can be reduced to find solutions of the famous Monge-Ampere equation too. Optimal transport theory is an important application of the Minkowski problem. This brief review paper introduced the optimal transportation problem and its applications at the end of this paper. At present, the Monge–Kantorovich optimal transport has found applications in wide range in different fields (including image registration and warping, reflector design, retrieving information from shadowgraphy and proton radiography, seismic tomography and reflection seismology, etc.). In conclusion, the author of this brief review paper described and studied the classical Minkowski problem in optimization languages. Minkowski duality is also needed to study furthermore. This is an interesting research topic for optimization related researchers. There is a wealth of information and insight in the classical Minkowski problem and its optimal transportation problem. This studying is worth the effort.},
     year = {2020}
    }
    

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    AB  - The research objective of this brief review paper is trying to introduce an optimization description for the classical (discrete) Minkowski problem to our mathematical readers, in the use of as plain and simple mathematical languages as possible. We also summarized some breakthroughs of the research history of classical Minkowski problem, and look forward to seeing a current breakthrough proposed. The classical Minkowski problem can be reduced to find solutions of the famous Monge-Ampere equation. The optimal transportation problem can be reduced to find solutions of the famous Monge-Ampere equation too. Optimal transport theory is an important application of the Minkowski problem. This brief review paper introduced the optimal transportation problem and its applications at the end of this paper. At present, the Monge–Kantorovich optimal transport has found applications in wide range in different fields (including image registration and warping, reflector design, retrieving information from shadowgraphy and proton radiography, seismic tomography and reflection seismology, etc.). In conclusion, the author of this brief review paper described and studied the classical Minkowski problem in optimization languages. Minkowski duality is also needed to study furthermore. This is an interesting research topic for optimization related researchers. There is a wealth of information and insight in the classical Minkowski problem and its optimal transportation problem. This studying is worth the effort.
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