Some Properties of the Volume Distance to Hypersurfaces

In this paper we consider the volume distance from a point to a convex hypersurface M ⊂ R, which is defined as the minimum (N + 1)-volume of a region bounded by M and a hyperplane through the point. We describe some of its properties, among them the centroid property, which says that any point is the centroid of its minimal section. We discuss the differentiability of the volume distance and show that it is smooth in a certain neighborhood of M . Besides, we obtain a formula for the hessian of the volume distance which makes possible to prove that the normalized hessian converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is given by a bilinear form associated with the shape operator of M . These convergence results provide a geometric interpretation of the Blaschke metric and the shape operator in terms of the volume distance. Mathematics Subject Classification (2010). 53A15.