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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 10, Pages 1657–1671 (Mi zvmmf239)

On the stability of inner and outer approximations of a convex compact set by a ball

S. I. Dudov, A. S. Dudova

Saratov State University, ul. Astrakhanskaya 42, Saratov, 410012, Russia

Abstract: The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.

Key words: estimation of a convex compact set by a ball, solution stability, outer and inner estimation, strong convexity.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:10, 1589–1602

Bibliographic databases:

UDC: 519.626

Citation: S. I. Dudov, A. S. Dudova, “On the stability of inner and outer approximations of a convex compact set by a ball”, Zh. Vychisl. Mat. Mat. Fiz., 47:10 (2007), 1657–1671; Comput. Math. Math. Phys., 47:10 (2007), 1589–1602

Citation in format AMSBIB
\Bibitem{DudDud07} \by S.~I.~Dudov, A.~S.~Dudova \paper On the stability of inner and outer approximations of a~convex compact set by a~ball \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 10 \pages 1657--1671 \mathnet{http://mi.mathnet.ru/zvmmf239} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2388618} \elib{https://elibrary.ru/item.asp?id=9535224} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 10 \pages 1589--1602 \crossref{https://doi.org/10.1134/S0965542507100028} \elib{https://elibrary.ru/item.asp?id=13561108} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-35648990143} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. I. Dudov, M. A. Osiptsev, “Stability of best approximation of a convex body by a ball of fixed radius”, Comput. Math. Math. Phys., 56:4 (2016), 525–540
2. M. V. Balashov, “Inscribed balls and their centers”, Comput. Math. Math. Phys., 57:12 (2017), 1899–1907
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