Packing and reflexivity in Banach spaces

Kottman, Clifford A.

doi:10.1090/S0002-9947-1970-0265918-7

Packing and reflexivity in Banach spaces
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by Clifford A. Kottman

Trans. Amer. Math. Soc. 150 (1970), 565-576

DOI: https://doi.org/10.1090/S0002-9947-1970-0265918-7

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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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