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

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l_3^p$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization

Basic Understanding of Condensed Phases of Matter via Packing Models

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Exact semidefinite programming bounds for packing problems

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.Comment: 24 page

kk-point semidefinite programming bounds for equiangular lines

We give a hierarchy of $k$-point bounds extending the Delsarte-Goethals-Seidel linear programming $2$-point bound and the Bachoc-Vallentin semidefinite programming $3$-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute~$4$, $5$, and $6$-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.Comment: 26 pages, 4 figures. New introduction and references update

Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling

The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give integrality gaps matching the best known approximation algorithms. For many other problems, however, ad-hoc techniques give better approximation ratios than \sos{} in the worst case, as shown by corresponding lower bound instances. Notably, in many cases these instances are invariant under the action of a large permutation group. This yields the question how symmetries in a formulation degrade the performance of the relaxation obtained by the \sos{} hierarchy. In this paper, we study this for the case of the minimum makespan problem on identical machines. Our first result is to show that $\Omega(n)$ rounds of \sos{} applied over the \emph{configuration linear program} yields an integrality gap of at least $1.0009$, where $n$ is the number of jobs. Our result is based on tools from representation theory of symmetric groups. Then, we consider the weaker \emph{assignment linear program} and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying $2^{\tilde{O}(1/\varepsilon^2)}$ rounds of the SA hierarchy to this stronger linear program reduces the integrality gap to $1+\varepsilon$, which yields a linear programming based polynomial time approximation scheme. Our results suggest that for this classical problem, symmetries were the main barrier preventing the \sos{}/ SA hierarchies to give relaxations of polynomial complexity with an integrality gap of~$1+\varepsilon$. We leave as an open question whether this phenomenon occurs for other symmetric problems