c-convexity
TODO: Brief definition, important examples (usual convex functions, nonexpansive functions)
Aka: Fenchel-Moreau conjugacy, abstract convexity
Literature
Jean-Jacques Moreau first noticed that the inner product in the Legendre-Fenchel transform could be replaced by a general coupling function:
- J.-J. Moreau, 1966: Fonctionnelles convexes (pdf)
- J.-J. Moreau, 1970: Inf-convolution, sous additivité, convexité des fonctions numériques
These papers are in French, so I cannot read them, but there are many secondary references:
- Rockafellar & Wets, 1998: Variational analysis, Sec 11.L: Generalized conjugacy
- Rubinov & Yang, 2003: Lagrange-type functions in constrained non-convex
optimization (doi), Sec 2.1.2: Fenchel-Moreau conjugacy and subdifferential
- Badly written, but cites all the standard books on abstract convexity, which are equally unreadable
Optimal transport
\(c\)-convexity has become a standard tool in optimal transport:
- Rachev & Ruschendorf, 1998: Mass transportation problems, Vol I, Sec 3.3
- Villani, 2003: Topics in optimal transportation, Sec 2.4
- Villani, 2009: Optimal transport: Old and new, Chapter 5
- Gangbo & McCann, 1996: The geometry of optimal transportation (doi), Appendices B-C