Volume distance to hypersurfaces: Asymptotic behavior of its hessian

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.