论文标题
在fejesTóth的猜想最大化器中，线之间的角度之和
On Fejes Tóth's conjectured maximizer for the sum of angles between lines
论文作者
论文摘要
通过$ {\ bf r}^{d+1} $选择$ n $无调线。如果这些线之间的分布在某些正顺式基础的坐标轴之间以$ {\ bf r}^{d+1} $分配，则将这些线之间的角度的总和猜想是最大化的。对于$ d \ ge 2 $，我们将猜想嵌入了一个参数的问题家族中，在这种问题中，我们试图最大程度地提高线条之间重新归一化角度的$α$ TH幂的总和。我们表明，猜想等于相同的配置成为所有$α> 1 $的{\ em unique}优化器（升至旋转）。我们在极限的最轻度排斥的限制情况下建立了主张的最优性和独特性。同样的结论扩展到$ n = \ infty $，只要我们假设只有有限的许多行是不同的。
Choose $N$ unoriented lines through the origin of ${\bf R}^{d+1}$. The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for ${\bf R}^{d+1}$. For $d \ge 2$ we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the $α$-th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the {\em unique} optimizer (up to rotations) for all $α>1$. We establish both the asserted optimality and uniqueness in the limiting case $α=\infty$ of mildest repulsion. The same conclusions extend to $N=\infty$, provided we assume only finitely many of the lines are distinct.
