Polytopes

Polytopes are geometric objects with flat sides that exist in any number of dimensions. In the most common mathematical usage, a polytope is the convex hull of a finite set of points in Euclidean space; equivalently, a convex polytope may be defined as a bounded intersection of finitely many half-spaces. When non-convex, the term polytope is sometimes used, though polyhedron is often reserved for three-dimensional cases.

In dimension d, a d-polytope has faces of every dimension from 0 up to d−1. The 0-faces are vertices, the 1-faces are edges, and the (d−1)-faces are facets. The collection of vertices and edges forms the 1-skeleton, a graph that encodes much of the polytope’s structure.

Two standard descriptions exist. A v-representation specifies the vertices directly, while an h-representation describes the polytope as a system of linear inequalities. The Minkowski–Weyl theorem states that every polytope can be described by either representation, and both descriptions are equivalent for convex polytopes.

Examples range from familiar polygons in the plane to three-dimensional polyhedra such as the tetrahedron, cube, and dodecahedron, and to higher-dimensional families. Important families include the simplex (generalization of a triangle/tetrahedron), the hypercube (n-cube), and the cross-polytope. Regular polytopes have symmetric, congruent faces and vertex figures; in two and three dimensions there are many regular examples, while in higher dimensions only three infinite families (simplex, hypercube, cross-polytope) are regular in general.

Historically, polytopes were studied from ancient geometry to modern theories developed by Schläfli and later by Coxeter. Polytopes underpin many areas of mathematics and applications, notably linear programming and polyhedral combinatorics, where polytopal structure describes feasible regions and their combinatorial properties.