On hotelling's formula for the volume of tubes and naiman's inequality

Two new derivations of the Hotelling-Naiman results on the volume of tubes about curves in spheres are presented. The first involves simple differential inequalities. The second is probabilistic, using the concept of upcrossing borrowed from the theory of Gaussian processes. The upcrossings method is extended to an harmonic regression problem not covered by the Hotelling-Naiman formulation.