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SPHERE COVERINGS, LATTICES, and TILINGS (In Low Dimensions)

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SPHERE COVERINGS, LATTICES, and TILINGS (In Low Dimensions) Technische Universitat¨ Munchen¨ Zentrum Mathematik SPHERE COVERINGS, LATTICES, AND TILINGS (in Low Dimensions) FRANK VALLENTIN Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Mathematik der Technischen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. PETER GRITZMANN Prufer¨ der Dissertation: 1. Univ.-Prof. Dr. JURGEN¨ RICHTER-GEBERT 2. Priv.-Doz. Dr. THORSTEN THEOBALD 3. o. Univ.-Prof. Dr. Dr. h.c. PETER MANFRED GRUBER Technische Universitat¨ Wien / Osterreich¨ (schriftliche Beurteilung) Prof. Dr. CHRISTINE BACHOC Universite´ de Bordeaux 1 / Frankreich (schriftliche Beurteilung) Die Dissertation wurde am 18. Juni 2003 bei der Technischen Universitat¨ Munchen¨ eingereicht und durch die Fakultat¨ fur¨ Mathematik am 26. November 2003 angenommen. Acknowledgements I want to thank everyone who helped me to prepare this thesis. This includes ALEXANDER BELOW,REGINA BISCHOFF,KLAUS BUCHNER,MATHIEU DUTOUR,CHRISTOPH ENGEL- HARDT,SLAVA GRISHUKHIN,JOHANN HARTL,VANESSA KRUMMECK,BERND STURMFELS, HERMANN VOGEL,JURGEN¨ WEBER. I would like to thank the research groups M10 and M9 at TU Munchen,¨ the Gremos at ETH Zurich,¨ and the Institute of Algebra and Geometry at Univer- sity of Dortmund for their cordial hospitality. Especially, I want to thank JURGEN¨ RICHTER-GEBERT for many interesting and helpful discus- sions, for being a constant source of motivation, and for showing me pieces and slices of his high-minded world of “Geometrie und Freiheit”; RUDOLF SCHARLAU for the kind introduction into the subject and for proposing this research topic; ACHILL SCHURMANN¨ for reading (too) many versions of the manuscript very carefully and for many, many discussions which give now the final(?) view on VORONO¨I’s reduction theory. Even more especially, I want to thank ANJA TRAFFAS for uncountable many things. Contents 1. Introduction 1 1.1. Dirichlet-Voronoi Polytopes . 1 1.2. Sphere Coverings and Sphere Packings . 2 1.3. Prerequisites . 4 1.4. Organisation of the Thesis . 4 2. VORONO¨I’s Reduction Theory 11 2.1. Delone Subdivisions . 12 2.2. Secondary Cones . 13 2.3. VORONO¨I’s Principal Domain of the First Type . 15 2.3.1. Selling Parameters . 16 2.3.2. Computation of the Secondary Cone . 17 2.3.3. SELLING’s Reduction Algorithm . 18 2.4. Bistellar Neighbours . 21 2.5. Main Theorem of VORONO¨I’s Reduction Theory . 23 2.6. Refinements and Sums . 25 2.7. Relations to the Theory of Secondary Polytopes . 27 3. Parallelohedra 29 3.1. Definition and Basic Properties . 29 3.2. VORONO¨I’s Conjecture . 30 3.3. Duality . 33 3.3.1. Definition . 33 3.3.2. Upper Bound Theorem . 34 3.3.3. Voronoi Vectors, Supporting Hyperplanes, and Facets . 34 3.3.4. Linearity and Rigidity . 35 3.4. Vonorms and Conorms . 36 3.5. Zonotopal Parallelohedra . 39 3.5.1. Definition and Basic Properties . 39 3.5.2. SEYMOUR’s Decomposition Theorem . 41 3.5.3. Delone Subdivisions, Dicings and Zonotopal Lattices . 44 4. Results in Low Dimensions 47 4.1. Dimension 1 . 48 4.2. Dimension 2 . 48 4.3. Dimension 3 . 48 4.4. Dimension 4 . 50 4.4.1. Four-Dimensional Delone Triangulations . 50 4.4.2. Vonorms and Conorms in Dimension 4 . 53 vi Contents 4.4.3. Towards a Classification . 54 4.4.4. Diagrams for Zonotopal Cases . 54 4.4.5. Diagrams for Non-Zonotopal Cases . 55 4.4.6. More Data . 61 4.5. Dimension 5 . 62 4.5.1. The Missing Delone Triangulation . 62 4.5.2. Rigid Forms in Dimension 5 . 63 4.6. Dimension 6 and Higher . 64 5. Determinant Maximization 69 5.1. The Determinant Maximization Problem . 69 5.2. Convexity of the Problem . 69 5.3. Relation to Semidefinite Programming . 70 5.4. Algorithms for the Determinant Maximization Problem . 71 6. Cayley-Menger Determinants 73 6.1. Definition and Basic Properties . 73 6.2. The Radius of the Circumsphere of a Simplex . 74 6.3. A Linear Matrix Inequality . 75 7. Solving the Lattice Covering Problem 79 7.1. A Restricted Lattice Covering Problem . 79 7.2. Interpretation of “Classical” Results . 82 8. Moments of Inertia 85 8.1. The Moment of Inertia and the Circumradius of a Simplex . 85 8.2. Linear Optimization on Equidiscriminant Surfaces . 87 8.3. A Local Lower Bound . 88 8.4. Applications . 88 8.4.1. The Lattice Covering of Ad∗ ......................... 88 8.4.2. Four-dimensional Lattice Coverings . 93 8.4.3. Invariants and Symmetry . 95 8.4.4. Navigating in the Graph of Delone Triangulations . 95 9. Results in Low Dimensions 99 9.1. Dimension 1 . 99 9.2. Dimension 2 . 99 9.3. Dimension 3 . 100 9.4. Dimension 4 . 100 9.5. Dimension 5 . 104 9.6. Dimension 6 . 105 9.6.1. The best known 6-dimensional lattice covering . 105 9.6.2. The second best known 6-dimensional lattice covering . 107 9.7. Dimension 7 . 108 9.8. Dimension 8 and Higher . 109 A. Glossary 111 A.1. Geometry of Numbers . 111 A.2. Polyhedra and Polytopes . 113 A.3. (Realizable) Oriented Matroids . 113 Chapter 1. Introduction The main subject of this thesis is the geometry of low dimensional lattices. We concentrate on two central problems: the lattice covering problem and the classification of polytopes that tile space by lattice translates. The lattice covering problem asks for the most economical way to cover d-dimensional space by equal, overlapping spheres whose centers form a lattice. Let us look at the two most prominent plane lattices together with their coverings. It is obvious that the covering which belongs to the square lattice is less economical than the one that belongs to the hexagonal lattice. The purpose of this introduction is two-fold. First, we define the rather intuitive notions we already used. We show how our future main actors, namely Dirichlet-Voronoi polytopes of lattices, come into the spotlight. Second, we give an outline of the thesis. 1.1. Dirichlet-Voronoi Polytopes A lattice L is a discrete subgroup of a d-dimensional Euclidean vector space (E; ( ; )). For any · · lattice there are always linearly independent vectors b1;:::; bn L so that they form a lattice 2 2 Chapter 1 Introduction basis, L = Zb1 Zbn. In the following we assume for simplicity and without loss of generality that n equals⊕ · · · ⊕d. The geometry of a lattice is encoded in its Dirichlet-Voronoi polytope. This is the polytope which contains all those points lying closer to the origin than to all other lattice points: DV(L; ( ; )) = x E : for all v L we have dist(x; 0) dist(x; v) : · · f 2 2 ≤ g If it is clear which scalar product we are using, we simply write DV(L). Dirichlet-Voronoi polytopes are very special polytopes. They tile E by translates of the form DV(L) + v, v L, in a face-to-face manner. They and all their facets are centrally symmetric. If one knows2 the Dirichlet-Voronoi polytope of a lattice it is easy to determine ge- ometrical information about the lattice. Its volume is the same as the lattice’s volume, i.e. the volume of a fundamental parallelotope. A fundamental parallelotope is given by a lattice basis i αibi : αi [0; 1] . The Dirichlet-Voronoi polytope’s circumsphere and the insphere give informationfP about2 coveringg respectively about packing properties. 1.2. Sphere Coverings and Sphere Packings d d A family of subsets = (Ki)i I of R , I a set of indices, is called a covering of R if each d K 2 d d point of R belongs to at least one of the sets Ki, i.e. R = i I Ki. A covering of R is a S 2 lattice covering if it is of the form (K +v)v L where L is a lattice. In summary, lattice coverings are those coverings which cover Rd by translated2 copies of a single body K and in addition the translates are the vectors of a lattice L. Although the general notion of density is very intuitive, it is not easy to give a good definition which exhibits all pathological cases (see e.g. [Kup2000]). But since we are interested only in the case of lattice coverings, pathological cases do not exist and we are fine with the following quite classical definitions. Rd r r d Let Cd(p; r) = x : maxi xi pi r = [ 2 ; 2 ] + p be a cube with side length r f 2 j − j ≤ g − d centered at p. We say that a collection = (Ki)i I of subsets of R has density ρ( ) if the limit K 2 K i I vol(Cd(0; r) Ki) lim P 2 \ r vol C (0; r) !1 d exists and if it equals ρ( ). Of course, there are cases where the limit does not exist and there are K even cases where the formula does not make any sense. But for lattice coverings (K+v)v L with 2 measurable body K there is no such problem as demonstrated in the first chapter of ROGERS’ little book [Rog1964]. There (Theorem 1.6) it is also shown that the density of a lattice covering vol K = (K + v)v L can be expressed as simple as ρ( ) = vol L where vol L is the volume of a fundamentalK parallelotope2 of the lattice L. K In the following we are only considering coverings consisting of solid spheres (balls). By d Bd(p; r) = x R : dist(x; p) r we denote the d-dimensional closed ball with center p f 2 ≤ g and radius r. A lattice L gives always a lattice covering of equal spheres (Bd(v; r))v L if the 2 radius r is large enough.
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