Ideal convergence of a sequence of Chebyshev radii of sets

In this paper, we investigate the diameters, Chebyshev radii, Chebyshev self-radii and inner radii of a sequence of sets in the normed spaces. We prove that if a sequence of sets is I -Hausdorff convergent to a set, the sequence of Chebyshev radii of that sequence is I-convergent. Similar relations are showed for the sequence of diameters, Chebyshev self-radii and inner radii of that sequence.

Keywords:

Chebyshev radius, Hausdorff convergence, I-convergence I-Hausdorff convergence,

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Amir, D., Chebyshev centers and uniform convexity, Pacific J. Math., 77(1) (1978), 1–6.
Castaing, C., Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
Fridy, J. A., On statistical convergence, Analysis, 5(4) (1985), 301–313. https://doi.org/10.1524/anly.1985.5.4.301
Fridy, J. A., Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187–1192. https://doi.org/10.1090/S0002-9939-1993-1181163-6
Hausdorff, F., Grundzüge der Mengenlehre, Chelsea Publishing Company, New York, 1949.
Kostyrko, P., Salat, T., Wilczynski, W., I-convergence, Real Anal. Exchange, 26(2) (2000/2001), 669–686.
Kuratowski, K., Topology. Vol. II, Academic Press, New York-London, 1968.
Mohiuddine, S. A., Hazarika, B., Alghamdi, M. A., Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33(14) (2019), 4549–4560. https://doi.org/10.2298/FIL1914549M
Mursaleen, M., Mohiuddine, S. A., On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62(1) (2012), 49–62. https://doi.org/10.2478/s12175-011-0071-9
Nuray, F., Rhoades, B. E., Statistical convergence of sequences of sets, Fasc. Math., 49 (2012), 87–99.
Papini, P. L., Wu, S., Nested sequences of sets, balls, Hausdorff convergence, Note Mat., 35(2) (2015), 99–114. https://doi.org/10.1285/i15900932v35n2p99
Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
Talo, Ö., Sever, Y., On Kuratowski I-convergence of sequences of closed sets, Filomat, 31(4) (2017), 899–912. https://doi.org/10.2298/FIL1704899T
Ward, J. D., Chebyshev centers in spaces of continuous functions, Pacific J. Math., 52 (1974), 283–287.