Volume Distance to Hypersurfaces: Asymptotic Behavior of its Hessian

TL;DR

This paper investigates the asymptotic behavior of the Hessian of the volume distance function to convex hypersurfaces, revealing its convergence to the affine Blaschke metric and relating it to the shape operator.

Contribution

It proves the convergence of the Hessian of the volume distance to the affine Blaschke metric and links this to the shape operator, providing geometric insights.

Findings

Hessian of volume distance converges to the affine Blaschke metric.

Rate of convergence is related to the shape operator.

Provides a geometric interpretation of the Blaschke metric.

Abstract

The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we obtain, after normalization, a symmetric bi-linear form Q. In this paper, we prove that Q 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.