Intro; Preface; Contents; The Tensorization Trick in Convex Geometry; 1 Introduction; 1.1 Tensor Power; 1.2 Tensor Powers and Polynomials; 1.3 Chebyshev Polynomials; 2 Packing Points in the Sphere; 2.1 Packing Points in the Sphere; 3 Approximating a Norm by a Polynomial and a Convex Body by an Algebraic Hypersurface; 3.1 Norms and Convex Bodies; 3.2 Some Geometric Corollaries; 3.3 Approximation by Convex Semi-algebraic Sets; 4 Approximation of Convex Bodies by Polytopes; 4.1 Convex Bodies and Polytopes; 4.2 Fine Approximations; 4.3 Coarse Approximations; 4.4 Intermediate Approximations

5 The Polynomial Method5.1 Constructing Neighborly Polytopes; 5.2 Bounding the Constant in the Grothendieck Inequality; 5.3 Polynomial Ham Sandwich Theorem; 5.4 Equiangular Lines; 5.5 A Counterexample to Borsuk's Conjecture; References; Contact Numbers for Sphere Packings; 1 Introduction; 2 Motivation from Materials Science; 3 Largest Contact Numbers in the Plane; 3.1 The Euclidean Plane; 3.2 Spherical and Hyperbolic Planes; 4 Largest Contact Numbers in 3-Space; 5 Empirical Approaches; 5.1 Contact Number Estimates for up to 11 Spheres; 5.2 Maximal Contact Rigid Clusters

6 Digital and Totally Separable Sphere Packings for d=2, 37 On Largest Contact Numbers in Higher Dimensional Spaces; 7.1 Packings by Translates of a Convex Body; 7.2 Contact Graphs of Unit Sphere Packings in mathbbEd; 7.3 Digital and Totally Separable Sphere Packings in mathbbEd; 8 Contact Graphs of Non-congruent Sphere Packings; References; The Topological Transversal Tverberg Theorem Plus Constraints; 1 Introduction; 2 Statement of the Main Results; 3 A Generalized Borsuk-Ulam Type Theorem and Two Lemmas; 3.1 Fadell-Husseini Index; 3.2 A Generalized Borsuk-Ulam Type Theorem

3.3 Two Lemmas4 Proofs; 4.1 Proof of the Topological Generalized Transversal Van Kampen-Flores Theorem; 4.2 Proof of the Topological Transversal Weak Colored Tverberg Theorem; References; On the Volume of Boolean Expressions of Balls
A Review of the Kneser-Poulsen Conjecture; 1 Introduction; 2 First Results; 3 Unions and Intersections of Balls in Spaces of Constant Curvature; 4 Jumping into Higher Dimensions
The Leapfrog Lemma; 5 Proof of the Kneser-Poulsen Conjecture in the Euclidean Plane; 6 Monotonicity of the Volume of Weighted Flowers

7 A Schläfli-Type Formula for Polytopes with Curved Faces7.1 Polytopes with Curved Faces in a Manifold and Their Variations; 7.2 Generalized Schläfli Formula in Einstein Manifolds; 8 Some Applications of the Generalized Schläfli Formula; 9 Application to a Problem of M. Kneser; 10 Alexander's Conjecture; 11 The Conjecture in More General Spaces; References; A Survey of Elekes-Rónyai-Type Problems; 1 The Elekes-Rónyai Problem; 1.1 Sums, Products, and Expanding Polynomials; 1.2 Extensions; 1.3 Applications; 1.4 About the Proof of Theorem1.1; 2 The Elekes-Szabó Problem