On the volume of tubular neighborhoods of real algebraic varieties

TL;DR

This paper derives bounds on the volume of tubular neighborhoods around real algebraic varieties, linking geometric properties to probabilistic analysis in numerical contexts.

Contribution

It provides a self-contained derivation of volume bounds for neighborhoods of algebraic sets, expressed via degrees of defining polynomials, including an unpublished result by Ocneanu.

Findings

Bounds on the probability of a random point being near an algebraic variety

Explicit relations between polynomial degrees and neighborhood volume

Application to probabilistic analysis of condition numbers

Abstract

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 The bounds are given in terms of the degrees of the defining polynomials, and contain as special case an unpublished result by Ocneanu.