Farthest points and strong convexity of sets
- G. E. Ivanov
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- E. Asplund, “Farthest points in reflexive locally uniformly rotund Banach spaces,” Israel J. Math. 4(4), 213–216 (1966). CrossRef
- M. Edelstein, “Fathest points of sets in uniformly convex Banach spaces,” Israel J. Math. 4(3), 171–176 (1966). CrossRef
- 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). CrossRef
- 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). CrossRef
- 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)].
- Farthest points and strong convexity of sets
Volume 87, Issue 3-4 , pp 355-366
- Cover Date
- Print ISSN
- Online ISSN
- SP MAIK Nauka/Interperiodica
- Additional Links
- optimization problem
- farthest points
- strong convexity of a set
- Banach space
- Fréchet differentiable norm
- Minkowski sum
- Hausdorff metric
- metric antiprojection
- Industry Sectors
- G. E. Ivanov (1)
- Author Affiliations
- 1. Moscow Institute of Physics and Technology, Moscow, Russia