Polytope

(1) A polyhedron.

(2) A geometric figure that is the union of a finite number of convex polyhedrons of an arbitrary number of dimensions arbitrarily arranged in n-dimensional space. This concept is often made use of in topology and can easily be extended to the case of n-dimensional space.

Let us consider a half space in the n-dimensional space Rⁿ— that is, the set of all points located on one side of some in — 1)-dimensional hyperplane of the space along with the points of the hyperplane itself. Analytically, the half space is the set of all points of Rⁿ whose coordinates satisfy a linear inequality of the form a₁x₁ + a₂x₂ + … + a_nx_n + b ≥0. The intersection of a finite number of half spaces—if it is bounded— is the most general convex polyhedron of arbitrary dimension ≤ n located in Rⁿ. A poly tope in the general sense of the word is the union of a finite number of such polyhedrons. When n = 2, we obtain two-dimensional polytopes, or polygons, which are not necessarily convex. One-dimensional polytopes are broken lines that need not be connected and may be branched—at any vertex any number of segments may meet. A zero-dimensional polytope is a finite set of points. A three-dimensional polytope can always be partitioned into polyhedrons of the simplest type —that is, into simplexes. Simplexes of dimension 0, 1, 2, and 3 correspond, respectively, to a point, a line segment, a triangle, and a tetrahedron, which is in general irregular. This partitioning, moreover, can be performed in such a way that either two of the resulting simplexes have no points in common or they share a face. Such partitions of a polytope into simplexes are called triangulations and constitute a fundamental research technique in combinatorial topology.

The concept of polytope permits of various generalizations. For example, curved polytopes are the images of polytopes under topological mappings; thus an arbitrary curved surface may be regarded as the topological image of a polyhedral surface. Another example is infinite polytope, which consists of an infinite set of convex polyhedrons (simplexes).