Abstract
For any subset A of the unit sphere of a Banach space X and for ε∈[0,2) the notion of ε-flatness is introduced as a “measure of non-flatness” of A. For any positive ε, construction of locally finite tilings of the unit sphere by ε-flat sets is carried out under suitable ε-renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.
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REFERENCES
Corson, H.H.: Collections of convex sets which cover a Banach space, Fund. Math. 49 (1961), 143–145.
Fonf, V.P.: Three characterizations of polyhedral Banach spaces, Ukrainian Math. J. 42 (1990), 1286–1290.
Fonf, V.P.: On the boundary of a polyhedral Banach space, Extracta Math. 15 (2000), 145–154.
Vesely, L.: Boundary of polyhedral spaces: an alternative proof, Extracta Math. 15 (2000), 213–217.
Contreras, M.D. and Payá, R.: On upper semicontinuity of duality mappings, Proc. Am. Math. Soc. 121 (1994), 451–459.
Cudia, D.: The geometry of Banach spaces. Smoothness, Trans. Am. Math. Soc. 110 (1964), 284–314.
Diestel, J.: Geometry of Banach spaces–Selected topics, Lecture Notes in Math., Vol. 485, Springer, New York, 1975.
Phelps, R.R.: Lectures on Choquet’s theorem, Van Nostrand Math. Studies 7 (1966).
Deville, R., Fonf, V.P. and Hajek, P.: Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998), 139–154.
Godefroy, G. and Zizler, V.: Roughness properties of norms on non-Asplund spaces, Michigan Math. J. 38 (1991), 461–466.
Simons, S.: A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708.
Fonf, V.P., Lindenstrauss, J., and Phelps, R.R.: Infinite Dimensional Convexity, North Holland Handbook on the Geometry of Banach spaces, Elsevier, Amsterdam, (2001).
Godefroy, G.: Some applications of Simons’ inequality, Serdica Math. J. 26 (2000), 59–78.
Negrepontis, S.: Banach spaces and topology, in Handbook of Set Theoretic Topology, Kunen, K. and Vaughan, J.E. (eds.), North-Holland, Amsterdam, 1984, 1045–1142.
Jiménez Sevilla, M. and Moreno, J.P.: Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997), 486–504.
Klee, V.: Do infinite-dimensional Banach spaces admit nice tilings?, Stud. Sci. Math. Ungar. 21 (1986), 415–427.
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Fonf, V., Zanco, C. Almost flat locally finite coverings of the sphere. Positivity 8, 269–281 (2004). https://doi.org/10.1007/s11117-004-5036-6
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DOI: https://doi.org/10.1007/s11117-004-5036-6