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

The Christoffel problem by fundamental solution of the Laplace equation