Characterization of Bounded Sets in Terms of Asymptotic Cones and Homotheties
Tóm tắt
A characterization of bounded sets in Banach spaces in terms of asymptotic cones and the Hausdorff deviations of sets from them homothetic images is obtained. Similar results for generalizations of the notion of boundedness are presented. Boundedness criteria have previously been known only for recessively compact sets.
Tài liệu tham khảo
R. T. Rockafellar, Convex Analysis (Princeton Univ., Princeton, NJ, 1970; Mir, Moscow, 1973).
A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities (Springer, New York, 2003).
D. T. Luc and J.-P. Penot, “Convergence of asymptotic directions,” Trans. Amer. Math. Soc. 353 (10), 4095–4121 (2001).
D. T. Luc, “Recessively compact sets: properties and uses,” Set-Valued Anal. 10 (1), 15–35(2002).
G. Köthe, Topological Vector Spaces. I (Springer-Verlag, Berlin, 1969).
C. Zălinescu, “Stability for a class of nonlinear optimization problems and applications,” in Nonsmooth Optimization and Related Topics, Ettore Majorana Internat. Sci. Ser. Phys. Sci. (Plenum, New York, 1989), Vol.43, pp. 437–458.
J.-P. Dedieu, “Critères de fermeture pour l’image d’un fermé non convexe par une multiapplication,” C. R. Acad. Sci. Paris Sér. A 287, 941–943 (1978).
D. T. Luc, Theory of Vector Optimization, in Lecture Notes in Econom. and Math. Systems (Springer-Verlag, New York, 1989), Vol. 319.
C. Zălinescu, “Recession cones and asymptotically compact sets,” J. Optim. Theory Appl. 77 (1), 209–220 (1993).
G. Isac and M. Théra, “Complementarity problem and the existence of the postcritical equilibrium state of the thin elastic plate,” in Seminaire d’Analyse Numérique (Univ. Paul Sabatier, Toulouse, 1985–86), pp.XI-1-XI-27.
G. C. Shephard and R. J. Webster, “Metrics for sets of convex bodies,” Mathematika 12, 73–88 (1965).
R. Bantegnie, “Convexité des hyperespaces,” Arch. Math. (Basel) 26 (4), 414–420 (1975).