On the volume of tubular neighborhoods of real algebraic varieties
Provides rigorous probabilistic bounds for tubular neighborhoods of real algebraic varieties, relevant to numerical analysis and condition number theory.
The paper derives bounds on the probability that a random point uniformly chosen from a ball lies within a given distance of a real algebraic variety, expressed in terms of the degrees of defining polynomials. The result generalizes an unpublished result by Ocneanu.
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 The bounds are given in terms of the degrees of the defining polynomials, and contain as special case an unpublished result by Ocneanu.