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In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes.
On a problem of Monge
L. V. Kantorovich · 1948
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Voronoi Diagrams
Herbert Edelsbrunner · 1987
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Polar factorization and monotone rearrangement of vector-valued functions
Yann Brenier · 1991
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Topics in optimal transportation
Cédric Villani · 2003
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Convex polyhedra Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky
A. D. Alexandrov · 2005
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Optimal transport: old and new
Cédric Villani · 2008
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From Knothe’s rearrangement to Brenier’s optimal transport map
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Stochastic optimization for large-scale optimal transport
Aude Genevay, Marco Cuturi, Gabriel Peyr’e, and Fancis Bach · 2016
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Variational principles for minkowski type problems, discrete optimal transport, and discrete monge-ampere equations
Xianfeng Gu, Feng Luo, Jian Sun, and Shing-Tung Yau · 2016
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Image-to-image translation with conditional adversarial networks
Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros · 2016
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Photo-realistic single image super-resolution using a generative adversarial network
Christian Ledig, Lucas Theis, Ferenc Huszár, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al · 2016
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Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros · 2016
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Guim Perarnau, Joost van de Weijer, Bogdan Raducanu, and Jose M Álvarez · 2016
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