Characterization of strict convexity for locally Lipschitz functions
It is well known that a smooth function \(f: X\to\mathbb{R}\) is convex iff \(f''(x;h, h)\geq 0\) for all \(x\in X\) and all \(h\in X\). The function \(f\) is strictly convex iff \(f''(x;h,h)> 0\) for ``almost'' all \(x\in X\) and all \(h\in X\). Here \[ f''(x;u,v)= \lim_{t\to 0} \frac{\langle\nabla f(x+ tu)-\nabla f(x),v\rangle}{t}, \] where \(\nabla f(x)\) is the symbol for the Gâteaux derivative of \(f\) at \(x\). In the paper, the author extends this result for nonsmooth, but locally Lipschitz functions using generalized directional derivatives of second order in which the gradient \(\nabla f\) is replaced by the Clarke subdifferential mapping \(\partial_C f\) and the limit is replaced by generalized limits. After the characterization of convexity for regularly locally Lipschitz functions, sufficient and necessary conditions for monotonicity and strict monotonicity of set-valued mappings \(F: X\rightrightarrows X^*\) are provided. Setting \(F=\partial_C f\), the author presents the main theorem of the paper using the fact that a real-valued function \(f\) is (strictly) convex if and only if the mapping \(\partial_C f\) is (strictly) monotone.
scientific article; zbMATH DE number 5652736
scientific article