Arbitrarily Complex Corner Polyhedra are Dense in $R^n $

Abstract

A cone determined by n independent linear inequalities in n-space is specified, for example by the constraint set of a mathematical programming problem. We show that under certain weak assumptions, the closed convex hull of the integer points in the cone has infinitely many extreme points. This result is used to prove the existence of small perturbations of the coefficients of any given cone, producing inequalities with integer coefficients, for which the orresponding closed convex hull of integer points has an arbitrarily large number of extreme points. This property may explain the volatility of convergence times of cutting plane algorithms.