 
Summary: A bound on kgonality of facets of the hypermetric
cone and related complexity problems
D.Avis V.P.Grishukhin
February 1992
Abstract
We give a bound on g h (n), the largest integer such that there is a g h (n)gonal
facet of the hypermetric cone Hyp n , g h (n) Ÿ 2 n\Gamma2 (n\Gamma1)! This proves simultaneously
the polyhedrality of the hypermetric cone. We give complete description of Delau
nay polytopes related to facets of Hyp n . We prove that the problem determining
hypermetricity lies in coNP and give some related NPhard problem.
1 Introduction
The hypermetric cone Hyp n of all hypermetrics on npoint set X is described by hyper
metric inequalities
X
1Ÿi!jŸn
b i b j d ij Ÿ 0 with
n
X
1
b i = 1; and b i 2 Z (1)
