Geometric permutations of non-overlapping unit balls revisited

Abstract : Given four congruent balls A, B, C, D in Rδ 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 counterexample of a highly degenerate nature (in the algebraic sense).
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Jae-Soon Ha, Otfried Cheong, Xavier Goaoc, Jungwoo Yang. Geometric permutations of non-overlapping unit balls revisited. Computational Geometry, Elsevier, 2016, 53, pp.36-50. ⟨10.1016/j.comgeo.2015.12.003⟩. ⟨hal-01393009⟩

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