  We prove that in any finite set of $\mathbb Z^d$ with $d\ge 3$, there is a
subset whose capacity and volume are both of the same order as the capacity of
the initial set. As an application we obtain estimates on the probability of
{\it covering uniformly} a finite set, and characterize some {\it folding}
events, under optimal hypotheses. For instance, knowing that a region of space
has an {\it atypically high occupation density} by some random walk, we show
that this random region is most likely ball-like
