Contents. Preface... VII. Part I Classical Topics Revisited

Transcription

1 Contents Preface VII Part I Classical Topics Revisited 1 Sphere Packings Kissing Numbers of Spheres One-Sided Kissing Numbers of Spheres On the Contact Numbers of Finite Sphere Packings Lower Bounds for the (Surface) Volume of Voronoi Cells in Sphere Packings On the Density of Sphere Packings in Spherical Containers Upper Bounds on Sphere Packings in High Dimensions Uniform Stability of Sphere Packings Finite Packings by Translates of Convex Bodies Hadwiger Numbers of Convex Bodies One-Sided Hadwiger Numbers of Convex Bodies Touching Numbers of Convex Bodies On the Number of Touching Pairs in Finite Packings Coverings by Homothetic Bodies - Illumination and Related Topics The Illumination Conjecture Equivalent Formulations The Illumination Conjecture in Dimension Three The Illumination Conjecture in High Dimensions On the X-Ray Number of Convex Bodies The Successive Illumination Numbers of Convex Bodies The Illumination and Covering Parameters of Convex Bodies On the Vertex Index of Convex Bodies

2 X Contents 4 Coverings by Planks and Cylinders Plank Theorems Covering Convex Bodies by Cylinders Covering Lattice Points by Hyperplanes On Some Strengthenings of the Plank Theorems of Ball and Bang On Partial Coverings by Planks: Bang s Theorem Revisited On the Volume of Finite Arrangements of Spheres The Conjecture of Kneser and Poulsen The Kneser Poulsen Conjecture for Continuous Contractions The Kneser Poulsen Conjecture in the Plane Non-Euclidean Kneser Poulsen-Type Results Alexander s Conjecture Densest Finite Sphere Packings Ball-Polyhedra as Intersections of Congruent Balls Disk-Polygons and Ball-Polyhedra Shortest Billiard Trajectories in Disk-Polygons Blaschke Lebesgue-Type Theorems for Disk-Polygons On the Steinitz Problem for Ball-Polyhedra On Global Rigidity of Ball-Polyhedra Separation and Support for Spindle Convex Sets Carathéodory- and Steinitz-Type Results Illumination of Ball-Polyhedra The Euler Poincaré Formula for Ball-Polyhedra Part II Selected Proofs 7 Selected Proofs on Sphere Packings Proof of Theorem A proof by estimating the surface area of unions of balls On the densest packing of congruent spherical caps of special radius Proof of Theorem The Voronoi star of a Voronoi cell in unit ball packings Estimating the volume of a Voronoi star from below Proof of Theorem Basic metric properties of Voronoi cells in unit ball packings Wedges of types I, II, and III, and truncated wedges of types I, and II The lemma of comparison and a characterization of regular polytopes

3 Contents XI Volume formulas for (truncated) wedges The integral representation of surface density in (truncated) wedges Truncation of wedges increases the surface density Maximum surface density in truncated wedges of type I An upper bound for the surface density in truncated wedges of type II The overall estimate of surface density in Voronoi cells Proof of Theorem The signed volume of convex polytopes The volume force of convex polytopes Critical volume condition Strictly locally volume expanding convex polytopes From critical volume condition and infinitesimal rigidity to uniform stability of sphere packings Selected Proofs on Finite Packings of Translates of Convex Bodies Proof of Theorem Monotonicity of a special integral function A proof by slicing via the Brunn Minkowski inequality Proof of Theorem Selected Proofs on Illumination and Related Topics Proof of Corollary Using Rogers Classical Theorem on Economical Coverings Proof of Theorem via the Gauss Map Proof of Theorem Using Antipodal Spherical Codes of Small Covering Radii Proofs of Theorem and Theorem From the Banach Mazur distance to the vertex index Calculating the vertex index of Euclidean balls in dimensions 2 and A lower bound for the vertex index using the Blaschke Santaló inequality and an inequality of Ball and Pajor An upper bound for the vertex index using a theorem of Rudelson Selected Proofs on Coverings by Planks and Cylinders Proof of Theorem On coverings of convex bodies by two planks A proof of the affine plank conjecture of Bang for non-overlapping cuts Proof of Theorem

4 XII Contents Covering ellipsoids by 1-codimensional cylinders Covering convex bodies by cylinders of given codimension Proof of Theorem Proof of Theorem Selected Proofs on the Kneser Poulsen Conjecture Proof of Theorem on the Monotonicity of Weighted Surface Volume Proof of Theorem on Weighted Surface and Codimension Two Volumes Proof of Theorem the Leapfrog Lemma Proof of Theorem The spherical leapfrog lemma Smooth contractions via Schläfli s differential formula Relating higher-dimensional spherical volumes to lower-dimensional ones Putting pieces together Proof of Theorem Monotonicity of the volume of hyperbolic simplices From Andreev s theorem to smooth one-parameter family of hyperbolic polyhedra Selected Proofs on Ball-Polyhedra Proof of Theorem Finite sets that cannot be translated into the interior of a convex body From generalized billiard trajectories to shortest ones Proofs of Theorems 6.6.1, 6.6.3, and Strict separation by spheres of radii at most one Characterizing spindle convex sets Separating spindle convex sets Proof of Theorem On the boundary of spindle convex hulls in terms of supporting spheres From the spherical Carathéodory theorem to an analogue for spindle convex hulls Proof of Theorem On the boundary of spindle convex hulls in terms of normal images On the Euclidean diameter of spindle convex hulls and normal images An upper bound for the illumination number based on a probabilistic approach

5 Contents XIII Schramm s lower bound for the proper measure of polars of sets of given diameter in spherical space An upper bound for the number of sets of given diameter that are needed to cover spherical space The final upper bound for the illumination number Proof of Theorem The CW-decomposition of the boundary of a standard ball-polyhedron On the number of generating balls of a standard ball-polyhedron Basic properties of face lattices of standard ball-polyhedra References

6