The equivalence of strict convexity and injectivity of the gradient in bounded level sets (Q1177236)

It is well-known that the notion of strict convexity of a real-valued function \(f: R^ n\to R\) can be characterized by strict monotony assumptions for the Clarke subdifferential set-mapping \(\partial f: R^ n\Rightarrow R^ n\). Since strict monotony implies injectivity the authors investigate whether injectivity of the mapping \(\partial f\) is also a sufficient condition for strict convexity of \(f\). For this, the authors regard Lipschitz functions over compact level sets. Assuming a rather weak regularity condition the equivalence of the following assertions can be proven: (1) \(f\) is strictly convex; (2) \(\partial f\) is injective; (3) the tangent planes to the epigraph of \(f\) generated by the subgradients are strictly supporting. Since the other implications are of classical nature the principal part of the paper is the proof of the implication (2) \(\rightarrow\) (3).