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Publication# On duality of canonical representations

2004

Report or working paper

Abstract

Every convex polyhedron in $R^d$ admits both H- and V-representations. In both cases, a representation is defined to be canonical if it is minimal and unique up to simple transformations. In general, canonical H- and V-representations are discussed separately, resulting in two different definitions. In contrast, the duality of polyhedral cones suggests a possible "unification" of the two types of canonical representations. In this paper, we describe a family of canonical representations, the dfnS-canonical representations, which definitions are the same for both H- and V-representation. We show that every S-canonical V-representation coincide with the S-canonical H-representation of a certain polyhedron. As a consequence, methods developed to determine S-canonical H-representations can be applied successfully in V-representation.

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Related concepts (6)

Related publications (3)

Polyhedron

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions.

Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . 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.

Convex polyhedra are important objects in various areas of mathematics and other disciplines. A fundamental result, known as Minkowski-Weyl theorem, states that every polyhedron admits two types of re

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.

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