RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2006, Volume 79, Issue 1, Pages 60–86 (Mi mz2674)

Weakly Convex Sets and Their Properties

G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.

DOI: https://doi.org/10.4213/mzm2674

Full text: PDF file (269 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2006, 79:1, 55–78

Bibliographic databases:

UDC: 517.982.252+517.978.2

Citation: G. E. Ivanov, “Weakly Convex Sets and Their Properties”, Mat. Zametki, 79:1 (2006), 60–86; Math. Notes, 79:1 (2006), 55–78

Citation in format AMSBIB
\Bibitem{Iva06} \by G.~E.~Ivanov \paper Weakly Convex Sets and Their Properties \jour Mat. Zametki \yr 2006 \vol 79 \issue 1 \pages 60--86 \mathnet{http://mi.mathnet.ru/mz2674} \crossref{https://doi.org/10.4213/mzm2674} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2252135} \zmath{https://zbmath.org/?q=an:1135.52301} \elib{http://elibrary.ru/item.asp?id=9210507} \transl \jour Math. Notes \yr 2006 \vol 79 \issue 1 \pages 55--78 \crossref{https://doi.org/10.1007/s11006-006-0005-y} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000235913800005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-31844437663} 

• http://mi.mathnet.ru/eng/mz2674
• https://doi.org/10.4213/mzm2674
• http://mi.mathnet.ru/eng/mz/v79/i1/p60

 SHARE:

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. G. E. Ivanov, M. V. Balashov, “Lipschitz continuous parametrizations of set-valued maps with weakly convex images”, Izv. Math., 71:6 (2007), 1123–1143
2. Balashov M.V. Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964
3. Goncharov V.V. Ivanov G.E., “Strong and Weak Convexity of Closed Sets in a Hilbert Space”, Operations Research, Engineering, and Cyber Security: Trends in Applied Mathematics and Technology, Springer Optimization and Its Applications, 113, ed. Daras N. Rassias T., Springer International Publishing Ag, 2017, 259–297
4. Lopushanski M.S., “Normal Regularity of Weakly Convex Sets in Asymmetric Normed Spaces”, J. Convex Anal., 25:3 (2018), 737–758
5. Ivanov G.E., Golubev M.O., “Strong and Weak Convexity in Nonlinear Differential Games”, IFAC PAPERSONLINE, 51:32 (2018), 13–18
6. V. N. Ushakov, A. A. Ershov, M. V. Pershakov, “Ob odnom dopolnenii k otsenke L.S. Pontryagina geometricheskoi raznosti mnozhestv na ploskosti”, Izv. IMI UdGU, 54 (2019), 63–73
•  Number of views: This page: 731 Full text: 230 References: 26 First page: 1