  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.
