Let V be a set of 2m (1 ≤ m < ∞) points in the plane. Two segments I, J with endpoints in V cross if relint I ∩ relint J is a singleton. A (perfect) cross-matching M on V is a set of m segments with endpoints in V such that every two segments in M cross. A halving line of V is a line l spanned by two points of V such that each one of the two open half planes bounded by l contains fewer than m points of V. Pach and Solymosi proved that if V is in general position, then V admits a perfect cross-matching iff V has exactly m halving lines. The aim of this note is to extend this result to the general case (where V is unrestricted).
@article{1228,
title = {Touching perfect matchings and halving lines},
journal = {ARS MATHEMATICA CONTEMPORANEA},
volume = {15},
year = {2018},
doi = {10.26493/1855-3974.1228.d7d},
language = {EN},
url = {http://dml.mathdoc.fr/item/1228}
}
Perles, Micha A.; Martini, Horst; Kupitz, Yaakov S. Touching perfect matchings and halving lines. ARS MATHEMATICA CONTEMPORANEA, Tome 15 (2018) . doi : 10.26493/1855-3974.1228.d7d. http://gdmltest.u-ga.fr/item/1228/