The volume of non-uniform tubular neighborhoods and an application to the -dimensional Szegِ theorem

Theorems of H. Weyl and J. Steiner regarding the volume of a tubular neighborhood about a hypersurface in R n are generalized to non-uniform tubular neighborhoods. The results provide new insight into the contribution of the geometry of the boundary of a region in R n to the coefficients of the asymptotics in the continuous n -dimensional version of Szegö’s theorem.