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Mat. Sb., 1996, Volume 187, Number 2, Pages 103–130 (Mi msb111)  

This article is cited in 46 scientific papers (total in 47 papers)

Strongly convex analysis

E. S. Polovinkin

Moscow Institute of Physics and Technology

Abstract: Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex $R$-hull of a set (the minimal strongly convex set containing the given set) is introduced; an explicit formula for the srongly convex $R$-hull of a set is obtained. The behaviour of the strongly convex $R$-hull under the variation of $R$ and of the sets is considered. An analogue of the Carathéodory theorem for strongly convex sets is obtained. The concept of a strongly extreme point is introduced, and a generalization of the Krein–Mil'man theorem for strongly convex sets is proved. Polyhedral approximations of convex and, in particular, of strongly convex compact sets are considered. Sharp error estimates for polyhedral and strongly convex approximations of such sets from inside and outside are established.


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English version:
Sbornik: Mathematics, 1996, 187:2, 259–286

Bibliographic databases:

UDC: 517.977
MSC: Primary 52A20, 52A27; Secondary 90D25
Received: 13.06.1995

Citation: E. S. Polovinkin, “Strongly convex analysis”, Mat. Sb., 187:2 (1996), 103–130; Sb. Math., 187:2 (1996), 259–286

Citation in format AMSBIB
\by E.~S.~Polovinkin
\paper Strongly convex analysis
\jour Mat. Sb.
\yr 1996
\vol 187
\issue 2
\pages 103--130
\jour Sb. Math.
\yr 1996
\vol 187
\issue 2
\pages 259--286

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