Geometric permutations of non-overlapping unit balls revisited

Given four congruent balls A,B,C,DA,B,C,D in RδRδ 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⩾7n⩾7. We also make a conjecture that would imply that n⩾4n⩾4 such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example that is algebraically highly degenerate.