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 Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 6, Pages 35–60 (Mi izv665)

Weak convexity in the senses of Vial and Efimov–Stechkin

G. E. Ivanov

Abstract: Research in convex analysis (in particular, in the theory of strongly convex sets developed in recent years) has made it possible to obtain important results in approximation theory, the theory of extremal problems, optimal control and differential game theory [1]–[3]. In many problems there arise non-convex sets that have weakened convexity properties, which enables one to study them using the methods of convex analysis. In this paper we study new properties of sets that are weakly convex in the sense of Vial or Efimov–Stechkin, that is, in the direct and dual senses. We establish relations between these two concepts of weak convexity. For subsets of Hilbert space that are weakly convex in the sense of Vial we prove a theorem on relative connectedness and a support principle.

DOI: https://doi.org/10.4213/im665

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English version:
Izvestiya: Mathematics, 2005, 69:6, 1113–1135

Bibliographic databases:

UDC: 517.982.252
MSC: 52A20, 52A27, 93B15, 91A23, 49N70, 49N75

Citation: G. E. Ivanov, “Weak convexity in the senses of Vial and Efimov–Stechkin”, Izv. RAN. Ser. Mat., 69:6 (2005), 35–60; Izv. Math., 69:6 (2005), 1113–1135

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv665
• https://doi.org/10.4213/im665
• http://mi.mathnet.ru/eng/izv/v69/i6/p35

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Balashov, G. E. Ivanov, “Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images”, Math. Notes, 80:4 (2006), 461–467
2. G. E. Ivanov, M. V. Balashov, “Lipschitz continuous parametrizations of set-valued maps with weakly convex images”, Izv. Math., 71:6 (2007), 1123–1143
3. Ivanov G.E., “Continuous selections of multifunctions with weakly convex values”, Topology Appl., 155:8 (2008), 851–857
4. Karasev R.N., “A measure of non-convexity in the plane and the Minkowski sum”, Discrete Comput. Geom., 44:3 (2010), 608–621
5. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$”, J. Math. Sci., 185:3 (2012), 360–366
6. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21–30
7. Balashov M.V. Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964
8. Salas D., Thibault L., “On Characterizations of Submanifolds Via Smoothness of the Distance Function in Hilbert Spaces”, J. Optim. Theory Appl., 182:1, SI (2019), 189–210
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