Volume of Tubes

Abstract

In Chapters 7 and 8we invested a good deal of time and energy in developing the many results we need from differential geometry. The time has now come to begin to reap the benefits of our investment, while at the same time developing some themes a little further for later exploitation. This chapter focuses on the celebrated volume-of-tubes formula of Wey1 [73, 168], which expresses the Lebesgue volume of a tube of radius ρ around a set M embedded in ℝ^l or S(ℝ{sl}) in terms of the radius of the tube1 and the Lipschitz–Killing curvatures of M (see Theorem 10.5.6). It is an interesting fact, particularly in view of the fact that² this is a book about probability that is claimed to have applications to statistics, and despite the fact thatWeyl’s formula is today the basis of a large literature in geometry, that the origins of the volume-of-tubes formulas were inspired by a statistical problem. This problem, along with its solution due to Hotelling [79], were related to regression analysis and involved the one-dimensional volume-of-tubes problem on a sphere, not unrelated to the computation we shall do in a moment.