Abstract
The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere \({{\mathbb S}^n}\). Necessary and sufficient conditions have been found by Firey (1967) and Berg (1969), by using the Green function of the Laplacian on the sphere. Expressing the Christoffel problem as the Laplace equation on the entire space ℝn+1, we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equation. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p ⩾ 2.
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Acknowledgements
The first author was supported by the One-Thousand-Young-Talents Program of China. The second author was supported by National Natural Science Foundation of China (Grant No. 11871345). The third author was supported by Australian Research Council (Grant Nos. DP170100929 and DP200101084).
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In Memory of Professor Zhengguo Bai (1916–2015)
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Li, QR., Wan, D. & Wang, XJ. The Christoffel problem by the fundamental solution of the Laplace equation. Sci. China Math. 64, 1599–1612 (2021). https://doi.org/10.1007/s11425-019-1674-1
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DOI: https://doi.org/10.1007/s11425-019-1674-1