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