User:MWinter4/Polytope theory

In mathematics polytope theory is the study of polytopes.

At the highest level polytope theory subdivides into the study of convex polytopes and the study of (potentially) non-convex polytopes. These two directions are distinct in the types of questions asked and techniques employed. While convex polytopes have an agreed upon definition, the right definition for general polytopes often dependes on the context and is more subtle.

In the modern context it is common for many authors to use the term "polytope" to refer to convex polytope only, and use polyhedral or polytopal surface for polytopes that are potentially non-convex.

Related to polytope theory, though usually seens as separate disciplines, is the study of abstract polytopes, infinite-dimensional polytopes, complex polytopes, etc. In contrast, the study of spherical polytopes and hyperbolic polytopes is traditionally seen as lying closer to polytope theory.

Classical polytope theory
Traditionally, the term "polytope theory" is used for the study of polytopes starting from dimension three and did not include, for example, the study if triangles. Classical polytope theory was then mainly concerned with polytopes of dimension exactly three.

The first polytopes studied in detail are the Platonic solids.

Results in classical polytope theory are
 * Euler's polyhedral formula
 * Cauchy's rigidity theorem
 * Steinitz' theorem, which characterizes edge graphs of 3-polytopes a 3-connected planar graphs.

Convex polytopes
Convex polytopes are convex bodies (compact convex sets) with a well-defined combinatorial structure. The study of convex polytopes therefore lies in the intersection of convex geometry and combinatorics.

Clasically, polytope theory deal mostly with polytopes in dimension up to three. The 19th century saw the inception of higher-dimensional geoemtry. Since then is is understood that polytopes in dimension $$d\ge 4$$ exhibit vastly different behavior (universality).

Moden convex polytope theory focuses mostly on polytopes in general dimensions, in particular, on dimension $$d\ge 4$$ where entirely new phenomena govern the combiantorics and geometry of convex polytopes.

Enumeration of combinatorial types

 * Gale diagrams
 * many simplicial polytopes, many neighborly polytopes

Realizations

 * Realization spaces and universality

Polyhedral combinatorics
Polyhedral combinatorics is one of the broadest and most active modern subdisciplines in polytope theory.


 * Euler-Poincaré equation and Dehn-Sommerville relations
 * upper bound theorem and g-theorem
 * oriented matroids

Distinction to simplicial spheres?

Famous open questions in polyhedral combiantorics include Kalai's $3^d$ conjecture.


 * Kalai's 3^d conjecture
 * Existence of a 4-polytope all whose facets are icodahedra
 * Is fatness unbounded?

Subdivisions and triangulations

 * polytopal complexes and shellings

Lattice polytopes and Erhart theory

 * lattice polytops
 * Erhart theory

Relations
Convex polytope theory relates to many subjects, including
 * linear programming
 * tropical geometry
 * toric varieties
 * commutative algebra

Non-convex polytopes
In modern terms, these are also known as polytopal surfaces or polyhedral surfaces.