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 /"-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

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... e property dual to this convexity condition is also exposed and examined. (Example 1.8).