Publication: Intersections of closed balls and geometry of Banach spaces
Loading...
Official URL
Full text at PDC
Publication Date
2004
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Universidad de Extremadura, Departamento de Matemáticas
Citations
Exportar
Abstract
In section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4 we discuss the interplay between porosity and MIP. Finally, in section 5 we are concerned with the stability of the (closure of the) sum of convex sets which are intersections of balls and
with Mazur spaces.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M.Acosta, M. Galan, New characterizations of the reflexivity in terms of the set of norm attaining functionals, Can. Math. Bull. 41(3)(1998), 279-289.
N. Aronszajn and P. Panitchpakdi, Extension of continuous tranformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439.
P. Bandyopadhyaya, A. Roy, Some stability results for Banach-spaces with the Mazur intersection property, Indagat. Math New Ser. 1(2)(1990), 137-154.
G. Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers, Dordrecht, 1993.
F. S. De Blasi, J, Myjak and P. L. Papini, Porous sets in best approximation theory, J. Lond. Math. Soc., 44 (1) (1991), 135–142.
R. Deville, Un théoreme de transfert pour la propriété des boules, Canad. Math. Bull. 30 (1987), 295–300.
R. Deville and J. Revalski, Porosity of ill-posed problems, Proc. Amer. Math. Soc., 128 (4) (2000), 1117—1124.
R. Deville,G. Godefroy, and V. Zizler, Smooth bump functions and geometry of Banach spaces, Mathematika 40(2)(1993), 305–321.
R.Deville,G.Godefroy,and V. Zizler,Smoothness and renormings in Banach spaces, vol. 64, Pitman Monograph and Surveys in Pure and Applied Mathematics, 1993.
Dongjian Chen and Bor-Luh Lin, Ball separation properties in Banach spaces,Rocky Mountain J.Math.28(1998), n. 3, 835–873.
Dongjian Chen and Bor-Luh Lin, On B-convex and Mazur sets of Banach spaces, Bull. Polish Acad. Sci.Math.43(3)(1995), 191–198.
G.A. Edgar, A long James space, Lecture notes in Math., vol. 794, 1979.
V.Fonf,On polyhedral Banach spaces,Math.Notes Acad.Sci.USSR, 45, 1989.
V. Fonf,Three characterizations of polyhedral Banach spaces, Ukrain. Math. J., 42, No. 9, 1990.
P. G. Georgiev, Mazur’s intersection property and a Krein-Milman type theorem for almost all closed, convex and bounded subsets of a Banach space, Proc. Amer. Math. Soc. 104 (1988), 157–164.
P. G.Georgiev, On the residuality of the set of norms having Mazur’s intersection property, Math-ematica Balkanica 5 (1991), 20–26.
P. G. Georgiev, A. S. Granero, M. Jiménez Sevilla and J. P. Moreno, Mazur intersection properties and differentiability of convex functions in Banach spaces,J.Lond. Math. Soc., (2) 61 (2000), 531–542.
J. R. Giles, D. A. Gregory and B. Sims, Characterization of normed linear spaces with Mazur’s intersection property, Bull. Austral. Math. Soc., 18 (1978), 471–476.
G. Godefroy and J. Kalton, The ball topology and its applications, Contemporary Math. 85 (1989), 195–238.
G. Godefroy, V. Montesinos and V. Zizler,Strong subdifferentiability of norms and geometry of Banach spaces, Comment. Math. Univ. Carolinae 36 (3) (1995), 417-421.
G. Godefroy, S. Troyanski, J. Whitfield and V. Zizler. Three–space problem for locally uniformly rotund renormings of Banach spaces, Proc. Amer. Math. Soc. 94 No. 4 (1985), 647–652.
A. S. Granero, M. Jiménez, A. Montesinos, J. P. Moreno, A. Plichko, On the Kunen-Shelah properties in Banach spaces, Studia Mathematica 157 (2) (2003), 97–120.
A. S. Granero, M. Jiménez Sevilla and J. P. Moreno Convex sets in Banach spaces and a problem of Rolewicz, Studia Math.1 29(1), (1998), 19–29.
A. S. Granero, J. P. Moreno and R. R. Phelps, Convex sets which are intersection of closed balls,Adv. in Math.,183 (1) (2004), 183–208.
P. M. Gruber, Baire categories in convexity, Handbook of convex geometry (P. M. Gruber and J. M. Wills eds.), North-Holland, 1993, 1327-1346.
P. M. Gruber, The space of convex bodies, Handbook of convex geometry (P. M. Gruber and J. M. Wills eds.), North-Holland, 1993, 301-318.
D. B. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89–108.
R. G. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), 261–268.
M. Jiménez Sevilla and J.P. Moreno, A note on norm attaining functionals, Proc. Amer. Math.Soc. 126 (1998), 1989–1997.
M. Jiménez Sevilla and J. P. Moreno, A note on porosity and the Mazur intersection property, Mathematika,47 (2000), 267–272.
M. Jiménez Sevilla and J. P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal., 144 (1997), 486–504.
J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (2)(1952), 323–326.
V.Klee,Polyhedral sections of convex bodies,Acta Mathematica 103 (1960), 243–267.
P. S. Kenderov and J. R. Giles, On the structure of Banach spaces with Mazur’s intersection property, Math. Ann. 291 (1991),463–473.
S. Negrepontis, Banach spaces and Topology, Handbook of set theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, 1984, 1045–1142
Kuratowski, Topology, vol. 1, Academic Press, 1966.
J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain ℓ1 and whose duals are not separable, Studia Math. 54 (1974), 81–105.
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, Springer-Verlag, 1979.
S. Mazur, ¨Uber schwache Konvergentz in den Raumen Lp, Studia Math. 4 (1933), 128–133.
J. P. Moreno, Geometry of Banach spaces with (α, ε)-property or (β, ε)-property, Rocky Mountain J. Math., 27 (1), (1997) 241-256.
J. P. Moreno, On the weak* Mazur intersection property and Fréchet differentiable norms on dense open sets, Bull. Sc. math., 122 (1998) 93-105.
J.R.Munkres, Topology, second edition, Prentice Hall (2000).
L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc.39 (1950), 28–46.
S. Negrepontis, Banach spaces and Topology, Handbook of set theoretic Topology (K.Kunen and J. E. Vaughan, eds.), North-Holland, 1984, 1045–1142
L. Narici and E. Beckenstein, The Hahn-Banach theorem: the life and times, Topology and its Applications 77 (1997) 193-211.
J. R. Partington, Equivalent norms on spaces of bounded functions , Isr. J. Math. 35(1980), No. 3, 205–209.
R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976–983.
R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Math. 1364, Springer Verlag, 1989; rev ed., 1993.
A. N. Plichko, A Banach space without a fundamental biorthogonal system, Soviet. Math. Dokl. 22 (1980) No. 2, 450–453.
D. Preiss and L. Zajicek, Stronger estimates of smallness of sets of Fr´echet nondifferentiability of convex functions, Proc. 11th Winter School, Suppl. Rend. Circ. Mat. di Palermo, Ser. II, nr. 3 (1984), 219-223.
M. Raja, On dual locally uniformly rotund norms, Israel J. Math. 129 (2002), 77-91.
A. Sersouri, The Mazur property for compact sets, Pacific J. Math. 133 (1988), no. 1, 185–195.
A. Sersouri, Mazur’s intersection property for finite-dimensional sets, Math. Ann. 283 (1989), no. 1, 165–170.
S. Shelah, Uncountable constructions for B. A., e.c. and Banach spaces, Isr. J. Math. 51 (1985), No. 4, 273–297.
R. C. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal. 13 (1989), 863-869.
F. Sullivan, Dentability, smoothability and stronger properties in Banach spaces, Indiana Math. J. 26 (1977), 545–553.
M. Talagrand, Renormages de quelques C(K), Israel J. Math. 54 (1986), 327–334.
S. L. Troyanski, On a property of the norm which is close to local uniform convexity, Math. Ann. 271 (1985), 305–313.
J. Vanderwerff, Mazur intersection properties for compact and weakly compact convex sets, Canad. Math. Bull. 41 (1998), no. 2, 225–230.
J. H. M. Whitfield and V. Zizler, Mazur’s intersection property of balls for compact convex sets, Bull. Austral. Math. Soc. 35 (1987), no. 2, 267–274.
J. H. M. Whitfield and V.Zizler,Uniform Mazur’s intersection property of balls, Canad. Math. Bull. 30 (1987), no. 4, 455–460.
L. Zajicek, Porosity and σ-porosity, Real Analysis Exchange 13 (1987-88), 314–350.
T. Zamfirescu, Porosity in convexity, Real Analysis Exchange 15 (1989-90), 424-436.
T. Zamfirescu, Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 1, 139–164.
V. Zizler, Renormings concerning the Mazur intersection property of balls for weakly compact convex sets, Math.Ann., 276, 1986, 61–66.