Abstract
Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding \({(a_0,x_0) \in A \times X}\) such that \({d(a_0,x_0) = \inf\left\{d(a,x) : a \in A, x \in X\right\}}\) (resp. \({d(a_0,x_0) = \sup\left\{d(a,x) : a \in A, x \in X\right\}}\)). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.
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References
Bridson M.R., Haefliger A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)
Bruhat F., Tits J.: Groupes réductifs sur un corps local. Données radicielles valuées I. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)
Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Am. Math. Soc., Providence RI (2001)
Cobzaş Ş.: Geometric properties of Banach spaces and the existence of nearest and farthest points. Abstr. Appl. Anal. 2005, 259–285 (2005)
Daneš J.: A geometric theorem useful in nonlinear functional analysis. Boll. Un. Mat. Ital. 6, 369–372 (1972)
De Blasi F.S., Myjak J.: On the minimum distance theorem to a closed convex set in a Banach space. Bull. Acad. Pol. Sci. Ser. Sci. Math. 29, 373–376 (1981)
De Blasi F.S., Myjak J., Papini P.L.: Porous sets in best approximation theory. J. Lond. Math. Soc. 44, 135–142 (1991)
De Blasi F.S., Myjak J., Papini P.L.: On mutually nearest and mutually furthest points of sets in Banach spaces. J. Approx. Theory 70, 142–155 (1992)
Dhompongsa S., Kirk W.A., Sims B.: Fixed points of uniformly lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)
Dontchev A., Zolezzi T.: Well-posed optimization problems, Lecture Notes in Math. 1543. Springer, Berlin (1993)
Espínola, R., Fernández-León, A., Pia̧tek, B.: Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity. Fixed Point Theory and Appl. 2010, Article ID 169837, 16 pages (2010)
Espínola R., Li C., López G.: Nearest and farthest points in spaces of curvature bounded below. J. Approx. Theory 162, 1364–1380 (2010)
Georgiev P.G.: The strong Ekeland variational principle, the strong drop theorem and applications. J. Math. Anal. Appl. 131, 1–21 (1988)
Gromov M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston (1999)
Hyers D.H., Isac G., Rassias Th.M.: Topics in Nonlinear Analysis and Applications. World Scientific Publishing Co., River Edge (1997)
Kaewcharoen, A., Kirk, W.A.: Proximinality in geodesic spaces. Abstr. Appl. Anal. 2006, Article ID 43591, 10 pages (2006)
Kirk W.A., Panyanak B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)
Kohlenbach U., Leustean L.: Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. J. Eur. Math. Soc. 12, 71–92 (2010)
Levitin E.S., Polyak B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Doklady 7, 764–767 (1966)
Li C.: On mutually nearest and mutually furthest points in reflexive Banach spaces. J. Approx. Theory 103, 1–17 (2000)
Penot J.P.: The Drop Theorem, the Petal Theorem and Ekeland’s Variational Principle. Nonlinear Anal. 10, 813–822 (1986)
Revalski J.P.: Generic properties concerning well-posed optimization problems. C. R. Acad. Bulgar. Sci. 38, 1431–1434 (1985)
Sababheh M., Khalil R.: Remotality of closed bounded convex sets in reflexive spaces. Num. Funct. Anal. Optim. 29, 1166–1170 (2008)
Stečkin S.B.: Approximation properties of sets in normed linear spaces (in Russian). Rev. Roum. Math. Pures Appl. 8, 5–18 (1963)
Zamfirescu T.: On the cut locus in Alexandrov spaces and applications to convex surfaces. Pacif. J. Math. 217, 375–386 (2004)
Zamfirescu T.: Extending Stechkin’s theorem and beyond. Abstr. Appl. Anal. 2005, 255–258 (2005)
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Espínola, R., Nicolae, A. Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces. Monatsh Math 165, 173–197 (2012). https://doi.org/10.1007/s00605-010-0266-0
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DOI: https://doi.org/10.1007/s00605-010-0266-0