## 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.

### MSC:

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

### Keywords:

Bouligand tangent cone; Clarke tangent cone; convexity
Full Text:

### References:

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