Subdifferential characterization of approximate convexity: The lower semicontinuous case (Q959933)

Having a Banach space \(X\) and a lower semicontinuous function \(f:X\to \mathbb{R}\cup\{+\infty\}\), the authors show as the main result of the paper that \(f\) is (directionally) approximately convex at \(x_0\in X\) if and only if \(\partial_Cf\) is (directionally) sub\-monotone at \(x_0\), where \(\partial_Cf\) denotes the Clarke subdifferential of \(f\). For the proof, the authors use a so called three points approximate mean value inequality stated and proved in the paper. The case of arbitrary subdifferentials is also discussed.