 
Summary: An Asymptotic Isoperimetric Inequality
Noga Alon
Ravi Boppana
Joel Spencer
Abstract
For a finite metric space V with a metric , let V n
be the metric space in which the distance
between (a1, . . . , an) and (b1, . . . , bn) is the sum
n
i=1 (ai, bi). 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
n.
1 The Main Results
Let V be a finite metric space with metric and with probability measure µ. On the set V n define
naturally the product probability measure
µn(a1, . . . , an) =
