Generalized submonotonicity and approximately convexity in Riemannian manifolds (Q2120264)

This article is concerned with the study of various important classes of non-smooth real-valued functions on Riemannian manifolds. More specifically, the authors investigate submonotone mappings, and lower-\(C^1\), semismooth, and approximately convex functions and as a result they generalize some statements from the Euclidean setting to Riemannian manifolds. It is shown that a locally Lipschitz function \(f\) is strictly approximately convex if and only if its Clarke subdifferential is strictly submonotone, which in turn is shown to be equivalent to \(f\) being lower-\(C^1\). In this context, it is moreover established that for a locally Lipschitz function satisfying a certain regularity condition semismoothness is equivalent to the submonotonicity of its Clarke subdifferential.