Geometric Packings of Non-Spherical Shapes

The focus of this thesis lies on geometric packings of non-spherical shapes in three-dimensional Euclidean space. Since computing the optimal packing density is difficult, we investigate lower and upper bounds for the optimal value. For this, we consider two special kinds of geometric packings: translative packings and lattice packings. We study upper bounds for the optimal packing density of translative packings. These are packings in which just translations and no rotations of the solids are allowed. Cohn and Elkies determined a linear program for the computation of such upper bounds that is defined by infinitely many inequalities optimizing over an infinite dimensional set. We relax this problem to a semidefinite problem with finitely many constraints, since this kind of problem is efficiently solvable in general. In our computation we consider three-dimensional convex bodies with tetrahedral or icosahedral symmetry. To obtain a program that is not too large for current solvers, we use invariant theory of finite pseudo-reflection groups to simplify the constraints. Since we solve this program by using numerical computations, the solutions might be slightly infeasible. Therefore, we verify the obtained solutions to ensure that they can be made feasible for the Cohn-Elkies program. With this approach we find new upper bounds for three-dimensional superballs, which are unit balls for the l^3_p norm, for p ∈ (1, ∞) \ {2} . Furthermore, using our approach, we improve Zong’s recent upper bound for the translative packing density of tetrahedra from 0.3840 . . . to 0.3683... , which is very close to the best known lower bound of 0.3673... The last part of this thesis deals with lattice packings of superballs. Lattice packing sare translative packings in which the centers of the solids form a lattice. Thus, any lattice packing density is in particular a lower bound for the optimal translative packing density. Using a theorem of Minkowski, we compute locally optimal lattice packings for superballs. We obtain lattice packings for p ∈ [1, 8] whose density is at least as high as the density of the currently best lattice packings provided by Jiao, Stillinger, and Torquato. For p ∈ (1, 2)\[log 2 3, 1.6], we even improve these lattice packings. The upper bounds for p ∈ [3, 8], as well as the numerical results for the upper bounds for p ∈ [1, log 2 , 3], are remarkably close to the lower bounds we obtain by these lattice packings