Abstract
We consider the existence and uniqueness of the farthest point of a given set A in a Banach space E from a given point x in the space E. It is assumed that A is a convex, closed, and bounded set in a uniformly convex Banach space E with Fréchet differentiable norm. It is shown that, for any point x sufficiently far from the set A, the point of the set A which is farthest from x exists, is unique, and depends continuously on the point x if and only if the set A in the Minkowski sum with some other set yields a ball. Moreover, the farthest (from x) point of the set A also depends continuously on the set A in the sense of the Hausdorff metric. If the norm ball of the space E is a generating set, these conditions on the set A are equivalent to its strong convexity.
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References
E. Asplund, “Farthest points in reflexive locally uniformly rotund Banach spaces,” Israel J. Math. 4(4), 213–216 (1966).
M. Edelstein, “Fathest points of sets in uniformly convex Banach spaces,” Israel J. Math. 4(3), 171–176 (1966).
N. V. Zhivkov, “Metric projections and antiprojections in strictly convex normed spaces,” C. R. Acad. Bulgare Sci. 31(4), 369–372 (1978).
S. Fitzpatrick, “Metric projections and the differentiability of distance functions,” Bull. Austral. Math. Soc. 22(2), 291–312 (1980).
M. V. Balashov and G. E. Ivanov, “On the farthest points of sets,” Mat. Zametki 80(2), 163–170 (2006) [Math. Notes 80 (1–2), 159–166 (2006)].
N. V. Efimov and S. B. Stechkin, “Some properties of Chebyshev sets,” Dokl. Akad. Nauk SSSR 118(1), 17–19 (1958).
L. P. Vlasov, “Approximative properties of sets in normed linear spaces,” Uspekhi Mat. Nauk 28(6), 3–66 (1973) [Russian Math. Surveys 28 (6), 1–66 (1973)].
V. S. Balaganskii and L. P. Vlasov, “The problem of convexity of Chebyshev sets,” Uspekhi Mat. Nauk 51(6), 125–188 (1996) [Russian Math. Surveys 51 (6), 1127–1190 (1996)].
J. M. Borwein, “Proximality and Chebyshev sets,” Optim. Lett. 1(1), 21–32 (2007).
A. R. Alimov and M. I. Karlov, “Sets with external Chebyshev layer,” Mat. Zametki 69(2), 303–307 (2001) [Math. Notes 69 (1–2), 269–273 (2001)].
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2007) [in Russian].
F.H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower-C 2 property,” J. Convex Anal. 2(1–2), 117–144 (1995).
F. Bernard, L. Thibault, and N. Zlateva, “Characterizations of prox-regular sets in uniformly convex Banach spaces,” J. Convex Anal. 13(3–4), 525–559 (2006).
J. Diestel, Geometry of Banach Spaces: Selected Topics, in Lecture Notes in Mathematics (Springer-Verlag, Berlin-Heidelberg-New York, 1975; Vishcha Shkola, Kiev, 1980), Vol. 485.
S. V. Konyagin, “On approximation properties of closed sets in Banach spaces and the characterization of strongly convex spaces,” Dokl. Akad. Nauk SSSR 251(2), 276–280 (1980) [Soviet Math. Dokl. 21, 418–422 (1980)].
S. B. Stechkin, “Approximation properties of sets in linear normed spaces,” in S. B. Stechkin, Selected Works in Mathematics (Fizmatlit, Moscow, 1998), pp. 270–281 [in Russian].
M. V. Balashov and G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces,” Izv. Ross. Akad. Nauk Ser. Mat. 73(3), 23–66 (2009) [Russian Acad. Sci. Izv. Math. 73 (3), 455–499 (2009)].
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Original Russian Text © G. E. Ivanov, 2010, published in Matematicheskie Zametki, 2010, Vol. 87, No. 3, pp. 382–395.
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Ivanov, G.E. Farthest points and strong convexity of sets. Math Notes 87, 355–366 (2010). https://doi.org/10.1134/S0001434610030065
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DOI: https://doi.org/10.1134/S0001434610030065