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# minFace -- computes the face of a Polyhedron or Cone where a weight attains its minimum

## Synopsis

• Usage:
F = minFace(w,P)
F = minFace(w,C)
• Inputs:
• w, , over ZZ or QQ with only one column representing a weight vector
• Outputs:
• F, Depending on the input, a Cone or a Polyhedron, the face where w attains its minimum

## Description

minFace computes the face of the given Polyhedron P or Cone C where w attains its minimum.
 i1 : P = hypercube 3 o1 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 3 number of facets => 6 number of rays => 0 number of vertices => 8 o1 : Polyhedron i2 : w = matrix {{1},{2},{0}} o2 = | 1 | | 2 | | 0 | 3 1 o2 : Matrix ZZ <--- ZZ i3 : F = minFace(w,P) o3 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 1 number of facets => 2 number of rays => 0 number of vertices => 2 o3 : Polyhedron i4 : vertices F o4 = | -1 -1 | | -1 -1 | | -1 1 | 3 2 o4 : Matrix QQ <--- QQ

## Ways to use minFace :

• "minFace(Matrix,Cone)"
• "minFace(Matrix,Polyhedron)"

## For the programmer

The object minFace is .