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The goal of the paper is to study the particular class of regularly${\mathcal{H}}$-convex functions, when ${\mathcal{H}}$ is the set${\mathcal{L}\widehat{C}}(X,{\mathbb{R}})$ of real-valued Lipschitz continuousclassically concave functions defined on a real normed space $X$. For anextended-real-valued function $f:X \mapsto \overline{\mathbb{R}}$ to be${\mathcal{L}\widehat{C}}$-convex it is necessary and sufficient that $f$ belower semicontinuous and bounded from below by a Lipschitz continuous function;moreover, each ${\mathcal{L}\widehat{C}}$-convex function is regularly${\mathcal{L}\widehat{C}}$-convex as well. We focus on${\mathcal{L}\widehat{C}}$-subdifferentiability of functions at a given point.We prove that the set of points at which an ${\mathcal{L}\widehat{C}}$-convexfunction is ${\mathcal{L}\widehat{C}}$-subdifferentiable is dense in itseffective domain. Using the subset ${\mathcal{L}\widehat{C}}_\theta$ of the set${\mathcal{L}\widehat{C}}$ consisting of such Lipschitz continuous concavefunctions that vanish at the origin we introduce the notions of${\mathcal{L}\widehat{C}}_\theta$-subgradient and${\mathcal{L}\widehat{C}}_\theta$-subdifferential of a function at a pointwhich generalize the corresponding notions of the classical convex analysis.Symmetric notions of abstract ${\mathcal{L}\widecheck{C}}$-concavity and${\mathcal{L}\widecheck{C}}$-superdifferentiability of functions where${\mathcal{L}\widecheck{C}}:= {\mathcal{L}\widecheck{C}}(X,{\mathbb{R}})$ isthe set of Lipschitz continuous convex functions are also considered. Someproperties and simple calculus rules for${\mathcal{L}\widehat{C}}_\theta$-subdifferentials as well as${\mathcal{L}\widehat{C}}_\theta$-subdifferential conditions for globalextremum points are established.

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions