Geometric permutations of non-overlapping unit balls revisited

Abstract

Given four congruent balls A, B, C, D in R-delta 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 >= 7. We also make a conjecture that would imply that n >= 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. (C) 2015 Elsevier B.V. All rights reserved.