Differences of two semiconvex functions on the real line

Abstract

It is proved that real functions on $\mathbb R$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $\mathbb R$ (i.e. those locally Lipschitz functions on $\mathbb R$ for which $f'_+(x) = \lim_{t \to x+} f'_+(t)$ and $f'_-(x) = \lim_{t \to x-} f'_-(t)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DSC_{\omega}$ of functions on $\mathbb R$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new notion of $[\omega]$-variation. We prove that $f \in DSC_{\omega}$ if and only if $f$ is continuous and there exists $D>0$ such that $f'_+$ has locally finite $[D \omega]$-variation. This result is proved via a generalization of the classical Jordan decomposition theorem which characterizes the differences of two $\omega$-nondecreasing functions (defined by the inequality $f(y) \geq f(x)- \omega(y-x)$ for $y>x$) on $[a,b]$ as functions with finite $[2\omega]$-variation. The research was motivated by a recent article by J. Duda and L. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.