scholarly journals Proximity theorems of discrete convex functions

2004 ◽  
Vol 99 (3) ◽  
pp. 539-562 ◽  
Author(s):  
Kazuo Murota ◽  
Akihisa Tamura
Keyword(s):  
Convex Functions ◽  
Discrete Convex

Cone superadditivity of discrete convex functions

2011 ◽  
Vol 135 (1-2) ◽  
pp. 25-44 ◽  
Author(s):  
Yusuke Kobayashi ◽  
Kazuo Murota ◽  
Robert Weismantel
Keyword(s):  
Convex Functions ◽  
Discrete Convex

scholarly journals Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth

1984 ◽  
Vol 36 (3) ◽  
pp. 569-576 ◽  
Author(s):  
Imre Bárány ◽  
Zoltán Füredi ◽  
János Pach
Keyword(s):  
Convex Functions ◽  
Circle Packing ◽  
Extremal Property ◽  
Discrete Convex ◽  
Image Position

A system of openly disjoint discs in the plane is 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, thenwhere r(C) denotes the radius of circle C, inf and sup are taken over all C ∊

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