On the volume of tubular neighbourhoods of real algebraic varieties

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 The bounds are given in terms of the degrees of the defining polynomials, and contain as special case an unpublished result by Ocneanu. Besides building a basis for general smoothed analysis results, this paper also serves to fill a gap in the literature by making available a complete and rigorous derivation of the real degree bounds used in previous work.