🔎Convex Geometry Unit 6 – Convex Polytopes and Their Properties

Key Concepts and Definitions

• Convex polytopes geometric objects formed by the intersection of a finite number of half-spaces in Euclidean space
• Vertices, edges, and faces fundamental building blocks of convex polytopes
  • Vertices points where two or more edges meet
  • Edges line segments connecting vertices
  • Faces polygonal regions bounded by edges
• Dimension of a polytope determined by the dimension of its affine hull (the smallest affine subspace containing the polytope)
• Simple polytopes polytopes where each vertex is incident to exactly $d$ edges, where $d$ is the dimension of the polytope
• Simplicial polytopes polytopes where all faces are simplices (triangles in 2D, tetrahedra in 3D)
• Dual polytopes obtained by interchanging the roles of vertices and faces in the original polytope (cube and octahedron)
• Polyhedral cone unbounded convex set formed by the intersection of a finite number of half-spaces with a common vertex

Historical Context and Development

• Euclid's Elements (300 BC) laid the foundation for the study of convex polytopes in ancient Greece
• Kepler's Mysterium Cosmographicum (1596) explored the relationship between Platonic solids and planetary orbits
• Euler's polyhedral formula (1758) established the relationship between the number of vertices, edges, and faces of a convex polyhedron: $V - E + F = 2$
• Cauchy's rigidity theorem (1813) proved that convex polyhedra are uniquely determined by their combinatorial structure (edge-vertex graph)
• Minkowski's Theorems (early 20th century) laid the groundwork for the modern theory of convex polytopes
  • Minkowski's Theorem states that every convex polytope is the Minkowski sum of its vertices
• Grünbaum's "Convex Polytopes" (1967) comprehensive treatise on the subject, consolidating and extending earlier results

Fundamental Theorems and Principles

• Minkowski-Weyl Theorem states that a set is a convex polytope if and only if it is the convex hull of a finite set of points or the intersection of a finite number of half-spaces
• Farkas' Lemma (1902) provides a characterization of the solvability of a system of linear inequalities
• Carathéodory's Theorem (1911) states that any point in the convex hull of a set $S$ can be expressed as a convex combination of at most $d+1$ points from $S$, where $d$ is the dimension of the affine hull of $S$
• Helly's Theorem (1923) if every subset of $d+1$ convex sets in $\mathbb{R}^d$ has a non-empty intersection, then the entire collection has a non-empty intersection
• Radon's Theorem (1921) any set of $d+2$ points in $\mathbb{R}^d$ can be partitioned into two disjoint subsets whose convex hulls intersect
• Brouwer's Fixed Point Theorem (1911) every continuous function from a convex compact set to itself has a fixed point

Geometric Properties of Convex Polytopes

• Combinatorial structure of a polytope captured by its face lattice, which encodes the incidence relations between faces of different dimensions
• Euler characteristic of a convex polytope $P$ is given by $\chi(P) = \sum_{i=0}^d (-1)^i f_i$, where $f_i$ is the number of $i$-dimensional faces
• Dehn-Sommerville equations relate the number of faces of different dimensions in a simplicial polytope
• Steinitz's Theorem (1922) a graph is the edge-vertex graph of a 3-dimensional convex polytope if and only if it is planar and 3-connected
• Balinski's Theorem (1961) the graph of a $d$-dimensional convex polytope is $d$-connected
• Gale diagrams provide a combinatorial representation of polytopes in terms of oriented matroids
• Schlegel diagrams allow for the visualization of higher-dimensional polytopes by projecting them onto a lower-dimensional space (4D polytope projected onto 3D space)

Algebraic Representation and Analysis

• Vertex representation of a convex polytope as the convex hull of a finite set of points: $P = \text{conv}(\{v_1, \ldots, v_n\})$
• Facet representation of a convex polytope as the intersection of a finite number of half-spaces: $P = \{x \in \mathbb{R}^d : Ax \leq b\}$
• Fourier-Motzkin elimination method for converting between vertex and facet representations by eliminating variables from a system of linear inequalities
• Ehrhart polynomial $i(P,t)$ counts the number of integer points in the $t$-th dilate of a convex polytope $P$
  • Ehrhart series $\sum_{t=0}^\infty i(P,t)x^t$ encodes information about the face numbers and volume of $P$
• Toric varieties algebraic varieties associated with rational convex polytopes, linking discrete geometry and algebraic geometry
• Stanley-Reisner ring algebraic object associated with a simplicial complex, capturing its combinatorial properties
• $f$-vector and $h$-vector sequences encoding the number of faces of each dimension in a convex polytope, related by the Dehn-Sommerville equations

Applications in Higher Dimensions

• Linear programming optimization problem of maximizing or minimizing a linear objective function subject to linear constraints, geometrically represented by a convex polytope
• Simplex method algorithm for solving linear programming problems by traversing the vertices of the feasible region polytope
• Cutting plane methods for integer programming, where the feasible region is successively approximated by intersecting it with half-spaces
• Voronoi diagrams partition of space into regions based on the nearest neighbor relation, with applications in computational geometry and data analysis (Delaunay triangulation)
• Convex hull algorithms for computing the smallest convex set containing a given set of points, with applications in computer graphics and robotics (QuickHull, Chan's algorithm)
• Polyhedral combinatorics study of the combinatorial properties of convex polytopes and their applications in optimization and discrete mathematics
• High-dimensional data analysis and machine learning, where data points are often represented as vectors in high-dimensional space, and convex polytopes are used for clustering and classification

Computational Aspects and Algorithms

• Complexity of convex hull computation depends on the dimension and the output representation (vertex or facet)
  • Incremental algorithms (e.g., beneath-beyond) add points one by one, updating the convex hull at each step
  • Divide-and-conquer algorithms (e.g., QuickHull) recursively partition the input points and merge the convex hulls of the subsets
• Vertex enumeration problem of listing all the vertices of a convex polytope given by its facet representation
  • Reverse search algorithm (Avis-Fukuda) explores the edge-vertex graph of the polytope using a depth-first search
• Facet enumeration problem of listing all the facets of a convex polytope given by its vertex representation
  • Double description method (Motzkin et al.) incrementally constructs the facet representation by adding vertices one by one
• Polytope volume computation can be done using triangulation methods or by integrating over the polytope's characteristic function
• Sampling from convex polytopes using random walks (e.g., hit-and-run) or Markov chain Monte Carlo methods
• Software libraries for convex polytopes, such as polymake, CGAL, and Sage, provide implementations of various algorithms and data structures

Advanced Topics and Current Research

• Ehrhart theory study of the integer points in dilates of convex polytopes, with connections to combinatorics and commutative algebra
  • Open problems include the characterization of Ehrhart polynomials and the classification of reflexive polytopes
• Gröbner bases and toric ideals algebraic tools for studying the integer points in convex polytopes and their applications in integer programming
• Polyhedral subdivisions and triangulations, such as regular and secondary polytopes, with applications in computational geometry and algebraic geometry
• Generalized permutohedra and associahedra classes of polytopes arising from combinatorial objects like permutations and trees, with connections to Hopf algebras and operads
• Polytope algebras and combinatorial Hopf algebras algebraic structures associated with convex polytopes, encoding their combinatorial and geometric properties
• Optimization over convex polytopes, including linear, integer, and semidefinite programming, with applications in operations research and engineering
• Convex polytopes in high-dimensional probability and statistics, such as the cross-polytope and the simplex, used in the study of concentration of measure phenomena