Journal of Convex Analysis 32 (2025), No. 2, 399--430
Copyright Heldermann Verlag 2025
60 Years of Cyclic Monotonicity: a Survey
Luigi De Pascale
Dip. di Matematica ed Informatica, Università degli Studi di Firenze, Firenze, Italy
luigi.depascale@unifi.it
Anna Kausamo
Dip. di Matematica ed Informatica, Università degli Studi di Firenze, Firenze, Italy
akausamo@gmail.com
Katarzyna Wyczesany
Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.
kwycz@cmu.edu
The primary purpose of this note is to provide an instructional summary of the state of the art regarding cyclic monotonicity and related notions. We will also present how these notions are tied to optimality in the optimal transport (or Monge-Kantorovich) problem, even though cyclic monotonicity is simply a property of a set and can be studied on its own merit.
Keywords: Monge–Kantorovich problem, optimality condition, c-cyclical monotonicity, Kantorovich potentials, transport map.
MSC: 49J45, 49N15, 49K21, 49K27, 49Q22.
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