Asian Journal of Mathematics

Volume 20 (2016)

Number 2

Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge–Ampère equations

Pages: 383 – 398



Xianfeng Gu (Department of Computer Science, Stony Brook University, New York, N.Y., U.S.A.)

Feng Luo (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Jian Sun (Department of Mathematics, Tsinghua University, Beijing, China)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge–Ampère equation (DMAE). A link between the discrete optimal transport, the discrete Monge–Ampère equation and the power diagram in computational geometry is established.


Monge–Ampère equation, Minkowski problem, Alexandrov problem, variational, power diagram

2010 Mathematics Subject Classification

Primary 52B55. Secondary 52B11, 65M99.

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Published 18 March 2016