How to Distribute N Points on the Surface of a Sphere ?

Volume of the Convex Hull

When has the convex hull of N points on the sphere volume as large as possible ?

For evaluation we have to use:

m: sum (l1)
D: volume of spherical ABC,
i.e. the volume of the tetrahedron defined by A,B,C and the sphere's center O.

The best polyhedron of a given combinatorical type fullfills a local criterion:

p = q/| q |   with   q = p1 × p2 + p2 × p3 + ... + pk × p1

The problem is completely solved for N = 4,..,8,12.
About the combinatorical type of N = 9,10,11,13,14 is known:

Literatur:

[1] Fejes-Tóth,Lászlo:
Lagerungen in der Ebene, auf der Kugel und im Raum
Springer, Berlin-Heidelberg, 1956, 1972
[2] Berman, J.D.; Hanes, K.:
Volumes of polyhedra inscribed in the unit sphere in E ³
Math.Ann 188 (1970), 78-84
[3] Hardin, R.H.; Sloane, N.J.A.; Smith, W.D.
www.research.att.com/~njas/maxvolumes/
[4] Wimmer, Lienhard:
Über das maximale Volumen der konvexen Hülle von Punkten auf der Einheitskugel
Dissertation, Salzburg, 1997

Lienhard Wimmer
2004-01-09