Geodesic
Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth
Canadian journal of mathematics, Tome 36 (1984) no. 3, pp. 569-576

Voir la notice de l'article provenant de la source Cambridge University Press

A system of openly disjoint discs in the planeis said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, then where r(C) denotes the radius of circle C, inf and sup are taken over all C ∊ 
Bárány, Imre; Füredi, Zoltán; Pach, János. Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth. Canadian journal of mathematics, Tome 36 (1984) no. 3, pp. 569-576. doi: 10.4153/CJM-1984-035-1
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