Sausage Conjecture

In $n$-D for $n\geq 5$ the arrangement of Hyperspheres whose Convex Hull has minimal Content is always a ``sausage'' (a set of Hyperspheres arranged with centers along a line), independent of the number of $n$-spheres. The Conjecture was proposed by Fejes Tóth, and solved for dimensions $\geq 42$ by Betke et al. (1994) and Betke and Henk (1998).

See also Content, Convex Hull, Hypersphere, Hypersphere Packing, Sphere Packing

References

Betke, U.; Henk, M.; and Wills, J. M. ``Finite and Infinite Packings.'' J. Reine Angew. Math. 453, 165-191, 1994.

Betke, U. and Henk, M. ``Finite Packings of Spheres.'' Discrete Comput. Geom. 19, 197-227, 1998.

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Problem D9 in Unsolved Problems in Geometry. New York: Springer-Verlag, 1991.

Fejes Tóth, L. ``Research Problems.'' Periodica Methematica Hungarica 6, 197-199, 1975.