Polyhedral combinatorics

Polyhedral combinatorics is a branch of combinatorial optimization that studies the properties and relationships of polyhedra, which are geometric structures defined by a finite number of linear inequalities. In the context of optimization, polyhedral combinatorics primarily focuses on the following aspects: 1. **Polyhedra and Convex Sets**: A polyhedron is a geometric figure in n-dimensional space defined by a finite number of linear inequalities.

Balinski's theorem is a result in the field of combinatorics and relates to the properties of convex polytopes. It states that every polytope in \( \mathbb{R}^d \) that is simple (meaning each vertex is the intersection of exactly \( d \) faces) can be decomposed into a fixed number of simplices (the simplest type of polytope, generalizing the concept of a triangle in higher dimensions).

A cyclic polytope is a specific type of convex polytope that arises in the context of combinatorial geometry and convex analysis. Defined for a given dimension and a set of points, cyclic polytopes have several interesting properties and applications in various fields, including algebraic geometry, optimization, and combinatorial mathematics.

The Dehn–Sommerville equations are a set of relationships in combinatorial geometry and convex geometry that relate the combinatorial properties of convex polytopes (or more generally, simplicial complexes) to their face counts. Specifically, these equations describe how the numbers of faces of different dimensions of a convex polytope are interconnected.

Eberhard's theorem is a result in the field of projective geometry, specifically concerning sets of points and their configurations. The theorem states that if a finite set \( S \) of points in the projective plane is such that every line intersects at least \( \lambda \) points of \( S \), then the total number of points in \( S \) is at most \( \lambda^2 \).

Euler's Gem refers to a remarkable relationship in geometry and topology, specifically relating to a formula known as Euler's formula for convex polyhedra.

The Euler characteristic is a topological invariant that describes a fundamental property of a topological space. It is often denoted by the Greek letter \( \chi \) (chi) and is defined for various types of spaces, including polyhedra, surfaces, and more generally, topological spaces.

Extension complexity is a concept from combinatorial optimization and theoretical computer science that deals with the complexity of representing convex sets and polytopes in terms of linear programming. Specifically, it studies how the size of a linear description (usually in terms of the number of constraints in the linear program) needed to define a convex set or polynomial can vary based on the way the set is extended or represented.

In geometry, a **facet** refers to a flat surface that forms part of the boundary of a higher-dimensional geometric object. The term is most commonly used in the context of polyhedra and polytopes. 1. **In Polyhedra**: For a three-dimensional polyhedron, a facet typically refers to one of the polygonal faces of the polyhedron. For example, a cube has six facets, all of which are square faces.

A Gale diagram, also known as a Gale's diagram or Gale's bipartite representation, is a graphical representation used in combinatorial optimization, particularly in the context of matching problems. In essence, a Gale diagram illustrates the relationships between two sets of items, typically referred to as agents and tasks, in a bipartite graph format. It facilitates visualization of the possible pairings between the two sets, often highlighting preferences or weights associated with each potential pairing.

The Hirsch conjecture is a famous statement in the field of computational geometry and polyhedral combinatorics. Proposed by the mathematician Warren Hirsch in 1957, the conjecture concerns the relationship between the dimensions of polyhedra and the lengths of their faces.

Kalai's 3-dimensional conjecture, proposed by Gil Kalai, pertains to the geometry of convex polytopes. The conjecture specifically addresses the conditions under which a simplicial complex can be realized as the nerve of a covering by open sets in a topological space. More concretely, it asserts that any simplicial complex that has a specific homotopy type will have a realization in a three-dimensional space.

"Lectures in Geometric Combinatorics" typically refers to instructional materials, texts, or courses focused on the intersection of combinatorics and geometry. This area of study explores various geometric structures using combinatorial methods and often involves topics such as: 1. **Convex Geometry**: The study of convex sets, convex polytopes, and their properties. This includes results related to the geometry of numbers, like Minkowski's theorem, and the relationships between different polytopes.

Linear programming relaxation is a technique used in optimization, particularly in the context of integer programming and combinatorial optimization problems. The primary goal of this technique is to simplify a complex integer programming problem into a linear programming one, which is generally easier to solve. Here's how it works: 1. **Original Problem**: Typically, an integer programming problem involves variables that are restricted to take only integer values (e.g., binary variables that can either be 0 or 1).

The Neighborly polytope is a specific type of convex polytope that is defined based on its combinatorial properties concerning its vertices.

The stable matching polytope is a geometric representation of the set of all stable matchings in a bipartite graph, where one set of vertices represents one group (such as men) and the other set represents another group (such as women). The concept is closely tied to the stable marriage problem, which seeks to find a stable match between two equally sized groups based on preferences.

Steinitz's theorem, named after mathematician Georg Cantor Steinitz, is a fundamental result in convex geometry and linear algebra related to the representation of convex polyhedra. The theorem states that a finite set of points in a Euclidean space forms the vertices of a convex polyhedron if and only if: 1. The points can be represented as the convex hull of those points.

"Unique sink orientation" is not a widely recognized term in literature or specific fields of study as of my last knowledge update in October 2023, and it may refer to a concept in a specialized area such as ecology, hydrology, or perhaps even computer science or data structures where "sink" (referring to a point where items are collected or processed) is a relevant term.