Power Series Expansions for Tube Volumes

Abstract

In this chapter we take a completely different approach to the study of volumes of tubes. We shall compute the first few terms of the power series of the volume function $V_P^M(r)$ as a function of r. In the same issue of the American Journal of Mathematics in which Weyl’s paper [Weyl1] appeared in 1939, there is an article [Ht] by the statistician Hotelling. In it Hotelling computes the first two nonzero terms of the expansion for $V_P^M(r)$ in the case that P is a curve in an arbitrary Riemannian manifold M. In fact, Weyl’s paper is partially a response to Hotelling’s paper. Hotelling discusses several applications of tube formulas to statistics.