再论非重叠单位球的几何排列
Geometric Permutations of Non-Overlapping Unit Balls Revisited
摘要：
给定$R^{d}$中的四个全等球$A, B, C, D$，它们具有不相交的内部，并且允许一条线以$ABCD$的顺序与它们相交，我们证明了连续球的中心之间的距离小于$A$和$D$的中心之间的距离。这使我们能够给出一个新的简短的证明，$n$相邻不相交全等球承认最多三个几何排列，两个，如果$n\ge 7$。我们还提出了一个猜想，这将意味着$n\geq 4$这样的球最多允许两个几何排列，并表明，如果猜想是假的，那么有一个反例的高度退化的性质。
英文摘要：
Given four congruent balls $A, B, C, D$ in $R^{d}$ that have disjoint interior and admit a line that intersects them in the order $ABCD$, we show that the distance between the centers of consecutive balls is smaller than the distance between the centers of $A$ and $D$. This allows us to give a new short proof that $n$ interior-disjoint congruent balls admit at most three geometric permutations, two if $n\ge 7$. We also make a conjecture that would imply that $n\geq 4$ such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example of a highly degenerate nature.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
