For any probability measure on the unit ball of an n-dimensional real Banach space, the expected distance between two independent samples is at most 2(1 - 2^{-n} f(n)), with f(n) ~ 2/(e n^2 log n).
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On averaged self-distances in finite dimensional Banach spaces
For any probability measure on the unit ball of an n-dimensional real Banach space, the expected distance between two independent samples is at most 2(1 - 2^{-n} f(n)), with f(n) ~ 2/(e n^2 log n).