Volumes of tubes, non polynomial behavior
Résumé
The behavior of the volume of the tube around a given compact subset in finite dimension, depending on the radius r, is an old and important question. It is related to many fields, like differential geometry, geometric measure theory, integral geometry, and also probability and statistics. Federer (Trans. Amer. Math. Soc. 93, 418–491, 1959), introduces the class of sets with positive reach, for which the volume is given by a polynomial in the radius 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 polynomial can be the volume of the tube, when the radius r tends to 0, 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- as introduced by Zähle (I. Math. Nachr. 119, 327–339, 1984).