Fenchel-Young inequality with a remainder and applications to convex duality and optimal transport

Guillaume Carlier (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

This short note is devoted to some applications of a simple quantitative form of the Fenchel-Young inequality in Hilbert spaces. Our initial motivation comes from a stability question in optimal transport. We derive from the quantitative form of the Fenchel-Young inequality a simple and constructive proof of the Brøndsted-Rockafellar theorem and a perturbed primal-dual attainment result in Hilbert spaces.

Mots clés: Fenchel-Young inequality in quantitative form; Brøndsted-Rockafellar theorem; Tilted convex duality; Stability of optimal transport

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier. Fenchel-Young inequality with a remainder and applications to convex duality and optimal transport. 2022. ⟨hal-03614052v2⟩