Mean Curvature and Asymptotic Volume of Small Balls

where Bp(t) is the ball of radius t centered at p and Bp (t) is the portion of the ball lying inside the sphere SR . Our goal is to show that, up to a negligible term, a similar formula holds for any hypersurface S in Rn, the factor 1/R being replaced with the mean curvature of the hypersurface. We briefly recall what mean curvature is. Let S be a hypersurface of class C2 in Euclidean n-space Rn, and assume that a unit normal vector field N : S → Rn has been chosen (this is always possible locally). It is a basic fact that the normal acceleration 〈c′′(t), N (c(t))〉 of a C2-curve c(t) on S depends only on its tangent vector V (t) = c′(t): indeed,