Strong and Weak Convexity of Sets and Functions

Abstract

In this paper we study two classes of sets, strongly and weakly convex sets. For each class we derive a series of properties which involve either the concept of supporting ball, an obvious extension of the concept of supporting hyperplane, or the normal cone to the set. We also study a class of functions, denoted ρ-convex, which satisfy for arbitrary points x1 and x2 and any value λ ∈ [0, 1] the classical inequality of convex functions up to a term ρ(1 − λ) λ‖x1 − x2‖2. Depending on the sign of the constant ρ the function is said to be strongly or weakly convex. We provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets. Finally, we give three applications: a separation theorem, a sufficient condition for global optimum of a nonconvex programming problem, and a sufficient geometrical condition for a set to be a manifold.