  We consider an extremal problem for subsets of high-dimensional spheres that
can be thought of as an extension of the classical isoperimetric problem on the
sphere. Let $A$ be a subset of the $(m-1)$-dimensional sphere
$\mathbb{S}^{m-1}$, and let $\mathbf{y}\in \mathbb{S}^{m-1}$ be a randomly
chosen point on the sphere. What is the measure of the intersection of the
$t$-neighborhood of the point $\mathbf{y}$ with the subset $A$? We show that
with high probability this intersection is approximately as large as the
intersection that would occur with high probability if $A$ were a spherical cap
of the same measure.
