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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2009 Volume 73, Issue 3, Pages 23–66 (Mi im2646)

This article is cited in 45 papers

Weakly convex and proximally smooth sets in Banach spaces

M. V. Balashov, G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: We establish interconnections between the conditions of weak convexity in the sense of Vial, weak convexity in the sense of Efimov–Stechkin, and proximal smoothness of sets in Banach spaces. We prove a theorem on the separation by a sphere of two disjoint sets, one of which is weakly convex in the sense of Vial and the other is strongly convex. We also prove that weakly convex and proximally smooth sets are locally connected, and study questions related to the preservation of the conditions of weak convexity and proximal smoothness under passage to the limit.

Keywords: proximal smoothness, weak convexity, uniform convexity, uniform smoothness, generating set, separation by a sphere, supporting ball.

UDC: 517.982.252

MSC: 46B07, 46B20, 52A30

Received: 04.04.2007

DOI: 10.4213/im2646


 English version:
Izvestiya: Mathematics, 2009, 73:3, 455–499

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