Renorming Banach spaces with the Mazur intersection property

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In this paper we give new sufficient and necessary conditions for a Banach space to be equivalently renormed with the Mazur intersection property. As a consequence, several examples and applications of these results are obtained. Among them, it is proved that every Banach space embeds isometrically into a Banach space with the Mazur intersection property, answering a question asked by Giles, Gregory, and Sims. We also prove that for every treeT, the spaceC0(T) admits a norm with the Mazur intersection property, solving a problem posed by Deville, Godefroy, and Zizler. Finally, assuming the continuum hypothesis, we find an example of an Asplund space admitting neither an equivalent norm with the above property nor a nicely smooth norm.
We thank G. Godefroy for Corollaries 2.8 and 4.4 as well as for many other helpful suggestions. We also thank J. Gomez and S. L. Troyanski for valuable discussions and comments.
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Dongjian Chen, Zhibao Hu, Bor-Luh Lin. Balls intersection properties of Banach spaces. Bull. Austral. Math. Soc., 45 (1992), pp. 333–342 M.D. Contreras, R. Payá. On upper semicontinuity of duality mappings. Proc. Amer. Math. Soc., 121 (1994), pp. 451–459 R. Deville. Un théorème de transfert pour la propriété des boules. Can. Math. Bull., 30 (1987), pp. 295–300 R. Deville, G. Godefroy, V. Zizler. Smoothness and Renormings in Banach Spaces, Pitman Monograph and Surveys in Pure and Applied Mathematics, Vol. 64 (1993) G.A. Edgar. A Long James Space, Lecture Notes in Math., Vol. 794 (1979) C. Finet. Renorming Banach spaces with many projections and smoothness properties. Math. Ann., 284 (1989), pp. 675–679 C. Finet, G. Godefroy. Biorthogonal systems and big quotient spaces. Contemporary Math., 85 (1989), pp. 87–110 C. Franchetti, R. Payá. Banach spaces with strongly subdifferentiable norm. Bolletino U.M.I.,7-B(1993),pp. 45–70 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), pp. 157–164 P.G. Georgiev. On the residuality of the set of norms having Mazur's intersection property. Math. Balkanica, 5 (1991), pp. 20–26 J.R.Giles, D.A. Gregory, B. Sims. Characterization of normed linear spaces with Mazur's intersection property. Bull. Austral. Math. Soc., 18 (1978), pp. 471–476 G. Godefroy, Nicely smooth Banach spaces, The University of Texas at Austin, Functional Analysis Seminar, 1984–1985 G. Godefroy. Asplund spaces and decomposable non separable Banach spaces.Rocky Mountain J.Math.,25(1995), pp. 1013–1024 G. Godefroy, J. Kalton. The ball topology and its applications. Contemporary Math., 85 (1989), pp. 195–238 G. Godefroy, V. Montesinos, V. Zizler. Strong subdifferentiability of norms and geometry of Banach spaces Comment. Math. Univ. Carolinae, 36 (1995), pp. 417–421 G. Godefroy, S. Troyanski, J. Whitfield, V. Zizler. Three-space problem for locally uniformly rotund renormings of Banach spaces.Proc. Amer. Math. Soc., 94 (1985), pp. 647–652 G. Godefroy, P.D. Saphar. Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math., 32 (1988), pp. 672–695 R.G. Haydon. A counterexample to several questions about scattered compact spaces. Bull. London Math. Soc., 22 (1990), pp. 261–268 R. G. Haydon, Trees in renormings M.P. Jiménez Sevilla. The Mazur intersection property and Asplund spaces- C.R. Acad. Sci. Paris, Série I, 321 (1995), pp. 1219–1223 M. Jiménez Sevilla, J.P. Moreno. On denseness of certain norms in Banach spaces.Bull. Austral. Math. Soc., 54 (1996), pp. 183–196 P.S. Kenderov, J.R. Giles. On the structure of Banach spaces with Mazur's intersection property. Math. Ann., 291 (1991), pp. 463–473 Bor-Luh, Lin, Wenyao, Zhang, Three-space property of Kadec–Klee renorming in Banach spaces J. Lindenstrauss, C. Stegall. Examples of separable spaces which do not contain ℓ1. Studia Math., 54 (1974), pp. 81–105 S.Mazur.Über schwache Konvergentz in den RaumenLp Studia Math., 4 (1933), pp. 128–133 A. Moltó,S.L. Troyanski. On uniformly Gâteaux differentiable norms in C(K). Mathematika, 41 (1994), pp. 233–238 J. P. Moreno, On the weak* Mazur intersection property and Fréchet differentiable norms on dense open sets, Bull. Sci. math. S. Negrepontis. Banach spaces and topology. Handbook of Set Theoretic TopologyNorth-Holland, Amsterdam (1984) J.R. Partington. Equivalent norms on spaces of bounded functions. Israel J. Math., 35 (1980), pp. 205–209 R.R. Phelps. A representation theorem for bounded convex sets. Proc. Am. Math. Soc., 11 (1960), pp. 976–983 S. Shelah. Uncountable constructions for B. A., e.c. and Banach spaces. Israel J. Math., 51 (1985), pp. 273–297 F. Sullivan. Dentability, smoothability and stronger properties in Banach spaces. Indiana Math. J., 26 (1977), pp. 545–553 M. Talagrand. Renormages de quelquesCK. Israel J. Math., 54 (1986), pp. 327–334 S. Troyanski. On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces. Studia Math., 39 (1971), pp. 173–180 V. Zizler. Renormings concerning the Mazur intersection property of balls for weakly compact convex sets. Math. Ann., 276 (1986), pp. 61–66