We study the problem of maximizing the minimal value over the sphere Sd-1⊂Rd\documentclass[12pt]{minimal}
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\begin{document}$$S^{d-1}\subset {\mathbb {R}}^d$$\end{document} of the potential generated by a configuration of d+1\documentclass[12pt]{minimal}
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\begin{document}$$d+1$$\end{document} points on Sd-1\documentclass[12pt]{minimal}
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\begin{document}$$S^{d-1}$$\end{document} (the maximal discrete polarization problem). The points interact via the potential given by a function f of the Euclidean distance squared, where f:[0,4]→(-∞,∞]\documentclass[12pt]{minimal}
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\begin{document}$$f:[0,4]\rightarrow (-\infty ,\infty ]$$\end{document} is continuous (in the extended sense), decreasing on [0, 4], and finite and convex on (0, 4], with a concave or convex derivative f′\documentclass[12pt]{minimal}
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\begin{document}$$f'$$\end{document}. We prove that the configuration of the vertices of a regular d-simplex inscribed in Sd-1\documentclass[12pt]{minimal}
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\begin{document}$$S^{d-1}$$\end{document} is optimal. This result is new for d>3\documentclass[12pt]{minimal}
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\begin{document}$$d>3$$\end{document} (certain special cases for d=2\documentclass[12pt]{minimal}
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\begin{document}$$d=2$$\end{document} and d=3\documentclass[12pt]{minimal}
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\begin{document}$$d=3$$\end{document} are also new). As a byproduct, we find a simpler proof for the known optimal covering property of the vertices of a regular d-simplex inscribed in Sd-1\documentclass[12pt]{minimal}
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\begin{document}$$S^{d-1}$$\end{document}.
