An Asymptotic Isoperimetric Inequality

Abstract.

For a finite metric space V with a metric $\rho$, let V ⁿ be the metric space in which the distance between (a ₁ , . . ., a _n ) and (b ₁ , . . ., b _n ) is the sum $\sum^{n}_{i = 1} \rho (a_i, b_i)$. We obtain an asymptotic formula for the logarithm of the maximum possible number of points in V ⁿ of distance at least d from a set of half the points of V ⁿ, when n tends to infinity and d satisfies $d \gg \sqrt {n}$.