By the properties of the Delaunay triangulation no triangle of the (spherical) convex hull has circumradius greater or equal the common radius. Thus we have:

m:	max (l∞)
D:	circumradius

This question is also known is "problem of friendly dictators". It had been solved for N=4...10,12, for N up to 20 an algorithm due to Tarnai und Gaspar yields the best results known.

The problem can be alternatively formulated as follows:

For a point-arrangement we study the smallest circumscribed ball of the tangential polyhedron with the same center as the sphere. Or we study the largest ball with the same center as the sphere inscribed the convex hull. Then we ask for the circumscribed ball with radius as small as possible resp. for the inscribed ball with radius as large as possible.

Using some vector-calculation it is easy to show these three problems to be identical.

Literatur:

[1] Fejes-Tóth,Gabor:

Kreisüberdeckungen der Sphäre
Stud.Sci.Math.Hung. 4 (1969), 225-247

[2] Fejes-Tóth,Lászlo:

Lagerungen in der Ebene, auf der Kugel und im Raum
Springer, Berlin-Heidelberg, 1956, 1972

[3] Hardin, R.H.; Sloane, N.J.A.; Smith, W.D.

www.research.att.com/~njas/coverings/

[4] Schütte, Kurt:

Überdeckung der Kugel mit höchstens acht Kreisen
Math.Ann 129 (1955), 181-186

[5] Tarnai, Tibor; Gáspár, Zsolt:

Covering a Sphere by Equal Circles the Rigidity of its Graph
Math. Proc. Camb. Phil. Soc. 110 (1991), 71-90

[6] Wimmer, Lienhard:

Covering the Sphere with 8 or 9 Equal Circles
preprint, 2003