In this talk, I will first present a very simple quantitative form of the Young-Fenchel inequality.  I will then discuss some applications: a short proof of the Brøndsted-Rockafellar in Hilbert spaces and a primal-dual attainment for perturbed convex minimization problems. I will finally explain how this inequality (or some generalizations) can be used for quantitative stability of optimal transport and entropic optimal transport (this part is based on a joint work with Paul Pegon and Luca Tamanini).

Zoom meeting: Link