Geometric permutations of non-overlapping unit balls revisited

Jae-Soon Ha (1) , Otfried Cheong (1) , Xavier Goaoc (2) , Jungwoo Yang (3)

1 KAIST - Department of Electrical Engineering [Korea Advanced Institute of Science and Technology]
2 LIGM - Laboratoire d'Informatique Gaspard-Monge
3 Aarhus University [Aarhus]

Given four congruent balls A, B, C, D in R^δ 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).

Mots clés: Transversal theory; Line transversal; Unit ball; Congruent balls; Geometric permutation

Domaines: Géométrie algorithmique [cs.CG]

Jae-Soon Ha, Otfried Cheong, Xavier Goaoc, Jungwoo Yang. Geometric permutations of non-overlapping unit balls revisited. Computational Geometry, 2016, 53, pp.36-50. ⟨10.1016/j.comgeo.2015.12.003⟩. ⟨hal-01393009⟩