In -D for 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 -spheres. The Conjecture was proposed by Fejes Tóth, and solved for dimensions 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.