Duality and subdifferential for convex functions on complete CAT(0) metric spaces

Abstract Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete C A T ( 0 ) (Hadamard) spaces. For a Hadamard space X , its dual is a metric space X ∗ which strictly separates non-empty, disjoint, convex closed subsets of X , provided that one of them is compact. If f : X → ( − ∞ , + ∞ ] is a proper, lower semicontinuous, convex function, then the subdifferential ∂ f : X ⇉ X ∗ is defined as a multivalued monotone operator such that, for any y ∈ X there exists some x ∈ X with x y ∈ ∂ f ( x ) . When X is a Hilbert space, it is a classical fact that R ( I + ∂ f ) = X . Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential ∂ ϵ f ( x ) is non-empty, for any ϵ > 0 and any x in efficient domain of f . Our results generalize duality and subdifferential of convex functions in Hilbert spaces.