论文标题
单纯性的高维球体上的极化问题
Polarization problem on a higher-dimensional sphere for a simplex
论文作者
论文摘要
我们研究了最大化球体上最小值的问题$ s^{d-1} \ subset \ mathbb r^d $，该构型生成的势能由$ s^{d-1} $上的$ d+1 $点（最大离散极化问题）产生。这些点是通过函数$ f $的欧几里得距离平方的函数的潜力相互作用的，其中$ f：[0,4] \ to（ - \ infty，\ infty] $是连续的（从通用意义上讲），并以$ [0,4] $和有限的$（0,4] $（0,4] $ conforiation $ f'$ f'的$ [0,4] $和有限的converiation $ f'。 $ d $ -simplex插入$ s^{d-1} $是最佳的。$ d> 3 $是新的（$ d = 2 $和$ d = 3 $的某些特殊情况，我们也是一个副产品，我们找到了更简单的证据。
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 $S^{d-1}$.
