Subdifferential set of the supremum of lower semi-continuous convex functions and the conical hull intersection property

In this paper we give some characterizations for the subdifferential set of the supremum of an arbitrary (possibly infinite) family of proper lower semi-continuous convex functions. This is achieved by means of formulae depending exclusively on the (exact) subdifferential sets and the normal cones to the domains of the involved functions. Our approach makes use of the concept of conical hull intersection property (CHIP, for short). It allows us to establish sufficient conditions guarantying explicit representations for this subdifferential set at any point of the effective domain of the supremum function.