Lipschitz continuity of convex functions (Q2232769)

The following characterization of the Lipschitz continuity of a convex function is provided. Theorem. Let \(f: X \rightarrow \overline{\mathbb R}\) be a proper lower semicontinuous convex function, \(l \geq 0\) and \(S\) be a nonempty open bounded set, \(\overline{S} \subset\mathrm{dom} f\). The function \(f\) is \(l\)-Lipschitz continuous (i.e., Lipschitz continuous with the Lipschitz constant \(l\)) on \(S\) iff \(f\) is locally \(l\)-Lipschitz with respect to \(S\) at every point in \(bd(S)\), i.e., for every \(x\in bd(S)\) there exists \(r > 0\) such that \(f\) is \(l\)-Lipschitz on \(S\cap B(x, r)\). This result is applied to extend a Lipschitz and convex function to the whole space and to establish the Lipschitz continuity of its Moreau envelope functions.