  The problem of determining the volume of a tubular neighbourhood has a long
and rich history. Bounds on the volume of neighbourhoods of algebraic sets have
turned out to play an important role in the probabilistic analysis of condition
numbers in numerical analysis. We present a self-contained derivation of bounds
on the probability that a random point, chosen uniformly from a ball, lies
within a given distance of a real algebraic variety of any codimension. The
bounds are given in terms of the degrees of the defining polynomials, and
contain as special case an unpublished result by Ocneanu.
