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Transactions of the American Mathematical Society

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Packing and reflexivity in Banach spaces


Author: Clifford A. Kottman
Journal: Trans. Amer. Math. Soc. 150 (1970), 565-576
MSC: Primary 46.10
DOI: https://doi.org/10.1090/S0002-9947-1970-0265918-7
MathSciNet review: 0265918
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Abstract: A measure of the ``massiveness'' of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the $ {l_p}$-spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to $ B$-convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.


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  • [1] M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313-317. MR 2, 221. MR 0003446 (2:221b)
  • [2] Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem and Pergamon Press, Oxford, 1961, pp. 123-160. MR 25 #2518. MR 0139079 (25:2518)
  • [3] Daniel P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc. 125 (1966), 114-146. MR 34 #4866. MR 0205031 (34:4866)
  • [4] R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 174-177. MR 13, 356. MR 0044024 (13:356d)
  • [5] -, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542-550. MR 30 #4139. MR 0173932 (30:4139)
  • [6] J. L. Kelley and I. Namioka, Linear topological spaces, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 29 #3851. MR 0166578 (29:3851)
  • [7] A. N. Kolmogorov and V. M. Tihomirov, $ \varepsilon $-entropy and $ \varepsilon $-capacity of sets in functional spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3-86; English transl., Amer. Math. Soc. Transl. (2) 17 (1961), 277-364. MR 22 #2890; MR 23 #A2031.
  • [8] Gottfried Köthe, Topologische lineare Räume. Vol. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin and New York, 1966. MR 33 #3069.
  • [9] Vlastimil Pták, Biorthogonal systems and reflexivity of Banach spaces, Czechoslovak Math. J. 9 (84) (1959), 319-326. MR 22 #891. MR 0110008 (22:891)
  • [10] Robert Whitley, The size of the unit sphere, Canad. J. Math. 20 (1968), 450-455. MR 37 #4576. MR 0228997 (37:4576)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0265918-7
Keywords: Packing, reflexivity, nearly isometric spaces, $ B$-convexity, $ P$-convexity, uniformly smooth, uniformly convex, geometry of the unit ball
Article copyright: © Copyright 1970 American Mathematical Society

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