A Complete Characterization of the Subdifferential Set of the Supremum of an Arbitrary Family of Convex Functions

Starting with some analysis of the support function of an arbitrary set, we obtain a formula for the subdifferential set of the supremum function of an arbitrary (possibly infinite) family of proper convex functions at each point of its effective domain, not necessarily at a continuity point. In this sense, our formula constitutes an extension of Theorem A of M. Volle ["Sous-différentiel d'une enveloppe supérieure de fonctions convexes", Comptes Rendus Acad. Sci. Paris I 317 (1993) 845-849], and also allows us to derive a generalization of a result A. Brøndsted ["On the subdifferential of the supremum of two convex functions", Math. Scand. 31 (1972) 225-230]. Our approach is based on a linearization via the Fenchel conjugate.