Tube volumes via functional equations

Abstract

Tube formulas for convex sets and for smooth submanifolds of Euclidean spaces have long been of interest for geometers. In the last two decades there has been great progress on tube formulas for badly behaved spaces such as fractals, sprays and strings. We give a new, simple proof of an interesting tube formula of M. Lapidus and E. Pearse for self-similar sprays using a natural functional equation satisfied by tube volumes of them.