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

This article is cited in 37 scientific papers (total in 38 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.

DOI: https://doi.org/10.4213/sm111

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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

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    2. S. I. Dudov, I. V. Zlatorunskaya, “Uniform estimate of a compact convex set by a ball in an arbitrary norm”, Sb. Math., 191:10 (2000), 1433–1458  mathnet  crossref  crossref  mathscinet  zmath  isi
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    4. M. V. Balashov, “An Analog of the Krein–Mil'man Theorem for Strongly Convex Hulls in Hilbert Space”, Math. Notes, 71:1 (2002), 34–38  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. M. V. Balashov, “On the $P$-Property of Compact Convex Sets”, Math. Notes, 71:3 (2002), 295–304  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    24. Mako J., Nikodem K., Pales Z., “On Strong (Alpha,F)-Convexity”, Math. Inequal. Appl., 15:2 (2012), 289–299  mathscinet  zmath  isi
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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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