The Christoffel problem by fundamental solution of the Laplace equation

TL;DR

This paper reformulates the Christoffel problem as a Laplace equation on the entire space, deriving new simpler conditions for its solvability and extending the analysis to the $L_p$ setting for $p \,\geq\, 2$.

Contribution

It introduces a novel approach using fundamental solutions of the Laplace equation to simplify the conditions for solving the Christoffel problem.

Findings

Derived new necessary and sufficient conditions for the Christoffel problem.

Extended the problem to the $L_p$ setting with sufficient conditions for $p \geq 2$.

Provided a simpler framework for analyzing convex solutions via Laplace equations.

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 $p\geq 2$.