Criteria for convexity of closed sets in Banach spaces. (English. Russian original) Zbl 1429.52001

Proc. Steklov Inst. Math. 304, 190-204 (2019); translation from Tr. Mat. Inst. Steklova 304, 205-220 (2019).
Let \(S\) be a closed subset of a Banach space \(\mathbb{X}\). For \(x \in X\), the notations \(T_S^B(x)\), \(T_S^C(x)\), \(N_S^C(x)\) and respectively \(N_S^P(x)\) designate the Bouligand tangent cone, the Clarke tangent cone, the Clarke normal cone and respectively the proximal normal cone to \(S\) at \(x\). The author proofs the equivalence of the following statements: (i) \(S\) is convex; (ii) for any \(x \in S\), \(S \subset x + T_S^B(x)\); (iii) for any \(x \in S\), \(S \subset x + T_S^C(x)\); (iv) for any \(x \in S\), \(S \subset x + \big(N_S^C(x)\big)^*\). If in addition, the space \(\mathbb{X}\) is uniformly convex, then the above statements are also equivalent with: (v) \(S \subset x + \big(N_S^P\big)^*(x)\). These criteria of convexity are then used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.


52A05 Convex sets without dimension restrictions (aspects of convex geometry)
26B25 Convexity of real functions of several variables, generalizations
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