高维球面上单形的极化问题
Polarization problem on a higher-dimensional sphere for a simplex
摘要：
本文研究了由$S^{d-1}$上的$d+1$点构成的势在球面$S^{d-1}\subset \mathbb R^d$上的最小值的最大化问题（最大离散极化问题）。这些点通过由欧几里得距离平方的函数$f$给出的势相互作用，其中$f:[0,4]\to (-\infty,\infty]$是连续的（在扩展意义上），并且在$[0,4]$上是递减的，并且在$(0,4]$上是有限的和凸的，具有凹或凸导数$f'$。证明了内接于$S^{d-1}$的正则$d$ -单形的顶点的构型是最优的.这个结果对于$d>3$是新的（对于$d=2$和$d=3$的某些特殊情况也是新的）。作为一个副产品，我们发现一个更简单的证明已知的最佳覆盖性质的正则$d$ -单形的顶点在$S^{d-1}$。
英文摘要：
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 $f:[0,4]\to (-\infty,\infty]$ is continuous (in the extended sense) and 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 $S^{d-1}$.
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