Polytope

A polytope is a geometric object with flat sides that exists in any number of dimensions, generalizing polygons in the plane and polyhedra in three dimensions. In Euclidean space R^n, a polytope can be defined as the convex hull of a finite set of points, equivalently as a bounded intersection of finitely many half-spaces. The two standard representations are the V-representation (through vertices) and the H-representation (through bounding half-spaces).

For a d-dimensional polytope, faces include vertices (0-faces), edges (1-faces), and facets (d−1-faces). The f-vector records the number of faces of each dimension. Polytopes can be regular or irregular; a regular polytope has high symmetry, with its automorphism group acting transitively on its flags. In two dimensions, this yields regular polygons; in three dimensions, the Platonic solids. Every polytope has a dual, in which vertices correspond to facets and vice versa.

Two important special classes are simple and simplicial polytopes. A simple d-polytope has exactly d edges meeting at every vertex; a simplicial d-polytope has every facet as a simplex. The convex case is the most common focus in mathematics, though non-convex polytopes are studied in broader contexts.

Polytopes play key roles in various fields: they describe feasible regions in linear and integer programming, model multi-dimensional combinatorial structures, and appear in computer graphics and crystallography. The study of polytopes connects geometry, combinatorics, and optimization, and has a rich history tracing back to Euler and Schläfli.