Conjugate convex functions and the epi-distance topology

Abstract:

Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a normed linear space, and let ${\Gamma ^ * }({X^ * })$ denote the proper, weak*-lower semicontinuous, convex functions on the dual ${X^ * }$ of $X$. It is well-known that the Young-Fenchel transform (conjugate operator) is bicontinuous when $X$ is reflexive and both $\Gamma (X)$ and ${\Gamma ^ * }({X^ * })$ are equipped with the topology of Mosco convergence. We show that without reflexivity, the transform is bicontinuous, provided we equip both $\Gamma (X)$ and ${\Gamma ^ * }({X^ * })$ with the (metrizable) epi-distance topology of Attouch and Wets. Convergence of a sequence of convex functions $\left \langle {{f_n}} \right \rangle$ to $f$ in this topology means uniform convergence on bounded subsets of the associated sequence of distance functional $\left \langle {d( \cdot ,{\text {epi}}{f_n})} \right \rangle$ to $d( \cdot ,{\text {epi}}f)$.