Geometric Permutations of Non-Overlapping Unit Balls Revisited

TL;DR

This paper investigates the geometric permutations of non-overlapping congruent balls in Euclidean space, providing new bounds and conjectures on the maximum number of permutations based on the arrangement of the balls.

Contribution

It offers a new short proof that at most three geometric permutations exist for disjoint congruent balls, and introduces a conjecture that could further reduce this maximum to two.

Findings

Proved that the distance between centers of consecutive balls in a permutation is less than the distance between the first and last.

Established that at most three geometric permutations exist for n interior-disjoint congruent balls, two if n ≥ 7.

Formulated a conjecture suggesting at most two permutations for n ≥ 4, with implications for potential counterexamples.

Abstract

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.