Subdifferential characterization of approximate convexity: the lower semicontinuous case

Throughout, X stands for a real Banach space, SX for its unit sphere, X ∗ for its topological dual, and 〈·, ·〉 for the duality pairing. All the functions f : X → R∪{+∞} are lower semicontinuous. The Clarke subdifferential , the Hadamard subdifferential and the Fréchet subdifferential of f are respectively denoted by ∂Cf , ∂Hf and ∂F f . The Zagrodny two points mean value inequality has proved to be a key tool. Here, we propose an approximate mean value inequality involving three points on a segment, which turns out to be well suited for establishing subdifferential criteria for approximate convexity.