Mean Value Representations and Curvatures of Compact Convex Hypersurfaces

It is shown that the kernels for mean value representations of points in R in terms of the integrals over piecewise smooth hypersurfaces are divergence free vector fields defined by homogeneous functions of degree −(n+1), whose restrictions to the unit sphere are positive and orthogonal to the first harmonics. By Minkowski problem, such a function is the reciprocal of the composition of the Gaussian curvature of a compact strictly convex hypersurface with its inverse Gauss map. For a compact strictly convex hypersurface, M, we define a dual surface M∗ via its support function. It is shown that the homogeneous extension of degree −(n+1) of the reciprocal of the composition of the curvature of M with its inverse Gauss map, as a function on the unit normal sphere, is equal to a homogeneous function defined by the curvature of M∗.