Volume of tubes, non polynomial behavior

TL;DR

This paper investigates the volume behavior of tubes around sets in geometric measure theory, demonstrating that even slight extensions of sets with positive reach can exhibit non-polynomial volume behavior.

Contribution

It provides an example showing how the volume of tubes can deviate from polynomial behavior for extended classes of sets beyond those with positive reach.

Findings

Volume of tubes can significantly deviate from polynomial form.

Extensions of sets with positive reach can exhibit non-polynomial tube volume behavior.

The example illustrates the limits of polynomial approximation in geometric measure theory.

Abstract

The behavior of the volume of the tube of distance r, around a given compact set M, is an old and important question with relations to many fields, like differential geometry, geometric measure theory, integral geometry, and also probability and statistics. Federer (1959) introduces the class of sets with positive reach, for which the volume is given by a polynom in r. For applications, in numerical analysis and statistics for example, an almost polynomial behavior is of equal interest. We exhibit an example showing how far to a polynom can be the volume of the tube, for the simplest extension of the class of sets with positive reach, namely the class of locally finite union of sets with positive reach satisfying a tangency condition introduced by Zahle (1984).