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Summary: On canonical representations of convex polyhedra
David Avis Komei Fukuda Stefano Picozzi
February 14, 2002
Abstract
Every convex polyhedron in the Euclidean space R d admits both H-representation and V-
representation. When working with convex polyhedra, in particular large-scale ones in high
dimensions, it is useful to have a canonical representation that is minimal and unique up to
some elementary operations. Such a representation allows one to compare two H-polyhedra or
two V-polyhedra eĆciently. In this paper, we de ne such representations that are simple and
can be computed in polynomial time. The key ingredients are redundancy removal for linear
inequality systems and aĆne transformations of polyhedra.
1 Introduction
A convex polyhedron or simply polyhedron in R d is the set of solutions to a nite system of inequal-
ities with real coeĆcients in d real variables. For a matrix A 2 R md and a vector b 2 R m , a pair
(b; A) is said to be an H-representation of a convex polyhedron P if P = fx 2 R d j b + Ax 0g.
Motzkin's decomposition theorem (see, e.g. [3, 4]) states that every polyhedron has another rep-
resentation called a V-representation. For matrices V 2 R pd and R 2 R qd , a pair (V; R) is said
to be a V-representation of a polyhedron P if P = conv(V ) + cone(R), where conv(M) (cone(M ),
respectively) denotes the convex hull (the nonnegative hull) of the row vectors of the matrix M .
In each of representations, there are trivial transformations that preserve the represented poly-
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