Convex polytope

Summary

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb{R}^n. Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. 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 identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to

Official source

https://en.wikipedia.org/wiki/Convex_polytope

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Every convex polyhedron in the Euclidean space

R^d

admits both H- and V-representation. For both formats, a representation is canonical if it is minimal and unique up to some elementary operations. In this paper, we extend the usual definition of canonical representation to a family of such representations that can be computed in polynomial time. In particular, this allows to define the lexico-smallest representation which computation is easy in practice. Furthermore, it guarantees certain sparsity property reflecting the real dimension of the studied object. As a consequence, H-representations of non-full dimensional polyhedra and V-representations of polyhedra without extreme points can be compared more efficiently.

2003

Convex polytope

The Lexico-smallest representation of convex polyhedra

On the duality of canonical representations rules for convex polyhedra

On the duality of canonical representations rules for convex polyhedra

The Lexico-smallest representation of convex polyhedra