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

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