  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 continuous
classically concave functions defined on a real normed space $X$. For an
extended-real-valued function $f:X \mapsto \overline{\mathbb{R}}$ to be
${\mathcal{L}\widehat{C}}$-convex it is necessary and sufficient that $f$ be
lower 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 ${\mathcal{L}\widehat{C}}$-convex
function is ${\mathcal{L}\widehat{C}}$-subdifferentiable is dense in its
effective domain. Using the subset ${\mathcal{L}\widehat{C}}_\theta$ of the set
${\mathcal{L}\widehat{C}}$ consisting of such Lipschitz continuous concave
functions 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 point
which 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
${\mathcal{L}\widecheck{C}}:= {\mathcal{L}\widecheck{C}}(X,{\mathbb{R}})$ is
the set of Lipschitz continuous convex functions are also considered. Some
properties and simple calculus rules for
${\mathcal{L}\widehat{C}}_\theta$-subdifferentials as well as
${\mathcal{L}\widehat{C}}_\theta$-subdifferential conditions for global
extremum points are established.
