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: Motivation by the question of testing for a nonlinear parameter in a regression model with independent, homoscedastic normal residuals, Hotelling (1939) was led to consider the geometric problem of computing the volume of a tube of given radius around a curve in s to the n-1 power the unit sphere in ir to the nth power. The answer involves only the arc length of the curve and not its curvature, providing the radius of the tube is sufficiently small that there is no self overlap in the tube. Starting from a somewhat different statistical setting Naiman (1986) arrived at the same geometric problem and showed that Hotelling's result (properly interpreted) is an upper bound for the volume of a tube of arbitrary radius. The purpose of this note is to give two new derivations of the Hotelling Naiman results. The first involves differential inequalities. The second is probabilistic, using the concept of upcrossing borrowed from the theory of Gaussian processes. In the context of Gaussian processes Knowles (1987) has observed that approximations obtained from Hotelling's result and bounds derived via upcrossings are related.

On Hotelling's Formula for the Volume of Tubes and Naiman's Inequality