We study the problem of maximizing the minimal value over the sphere \(S^{d-1}\subset {\mathbb {R}}^d\) of the potential generated by a configuration of \(d+1\) points on \(S^{d-1}\) (the maximal discrete polarization problem). The points interact via the potential given by a function f of the Euclidean distance squared, where \(f:[0,4]\rightarrow (-\infty ,\infty ]\) is continuous (in the extended sense), decreasing on [0, 4], and finite and convex on (0, 4], with a concave or convex derivative \(f'\). We prove that the configuration of the vertices of a regular d-simplex inscribed in \(S^{d-1}\) is optimal. This result is new for \(d>3\) (certain special cases for \(d=2\) and \(d=3\) 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 \(S^{d-1}\).

我们研究了在由\(d+1\)点的配置产生的势的球体\(S^{d-1}\subset {\mathbb {R}}^d\)上最大化最小值的问题在 \(S^{d-1}\)（最大离散极化问题）上。这些点通过欧几里德距离平方函数f给出的势相互作用，其中\(f:[0,4]\rightarrow (-\infty ,\infty ]\)是连续的（在扩展意义上），在[0,4]，和有限和凸对（0,4]，具有凹面或凸面衍生物 \（F'\） ，证明了一个普通的顶点的配置d单纯形内切在\（S ^ {d-1}\)是最优的。这个结果对于\(d>3\) 来说是新的（\(d=2\)和\(d=3\) 的某些特殊情况也是新的）。作为副产品，我们找到了内接于\(S^{d-1}\)的正则d -单纯形 顶点的已知最优覆盖属性的更简单证明。