The Christoffel problem by fundamental solution of the Laplace equation

Abstract

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere $S^n$. Necessary and sufficient conditions have been found by Firey and Berg, using the Green function of the Laplacian on the sphere. Expressing the Christoffel problem as the Laplace equation on the entire space $R^{n+1}$, we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equations. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the $L_p$ extension of the Christoffel problem and provide sufficient conditions for the problem, for the case $p\geq 2$.