Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms 'convex polytope' and 'convex polyhedron' interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms 'convex polytope' and 'convex polyhedron' interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set, while others (including this article) allow polytopes to be unbounded. The terms 'bounded/unbounded convex polytope' will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n-1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum. In 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaum's book, and in some other texts in discrete geometry, convex polytopes are often simply called 'polytopes'. Grünbaum points out that this is solely to avoid the endless repetition of the word 'convex', and that the discussion should throughout be understood as applying only to the convex variety. A polytope is called full-dimensional if it is an n-dimensional object in Rn. A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaum's definition is in terms of a convex set of points in space. Other important definitions are: as the intersection of half-spaces (half-space representation) and as the convex hull of a set of points (vertex representation). In his book Convex polytopes, Grünbaum defines a convex polytope as a compact convex set with a finite number of extreme points: This is equivalent to defining a bounded convex polytope as the convex hull of a finite set of points, where the finite set must contain the set of extreme points of the polytope. Such a definition is called a vertex representation (V-representation or V-description). For a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope.