• An m-hemi-metric d is a function defined over all (m+1)-uples of {1,...,n} such that
• d(x_1,....,x_[m+1]) is invariant by permutation of the x_i
• d(x_1,....,x_[m+1])=0 if two x_i are identical
• d(x_1,....,x_[m+1])>=0
• d(x_1,....,x_[m+1])<=sum_{i=1}^{m+1} d(x_1,....,x_[i-1},x_[i+1],...,x_[m+1])
• The m-hemi-metric cone HMET^m_n is the polyhedral cone of all hemi-metrics
• Given an unordered (m+1)-partition $S_{1}, \ldots, S_{m+1}$ of {1,...,n}, the m-hemi-metric, which we denote \alpha(S_{1}, ...., S_{m+1}), and call m-partition hemi-metric is defined by
• d(x_1,....,x_[m+1])=0 if two x_i belong to the same subset
• d(x_1,....,x_[m+1])=1 otherwise
• The m-hemi-cut cone HCUT^m_n is the positive span of all m-partition-hemi-metrics

HCUT^2_5

dimension	10
group	Sym(5)
extreme rays	25 in two orbits
facets	120 in four orbits

see files 2-P5.tex and 2-P5.ps for details
 

HMET^2_5

dimension	10
group	Sym(5)
extreme rays	37 in three orbits
facets	30 in two orbits

see files 2HMET5.tex and 2HMET5.ps for details

HCUT^3_6

dimension	15
group	Sym(6)
extreme rays	65 in two orbits
facets	4065 in 16 orbits

see files 3-P6.tex and 3-P6.ps for details

HMET^3_6

dimension	15
group	Sym(6)
extreme rays	287 in 5 orbits
facets	45 in two orbits

see files 3HMET6.tex and 3HMET6.ps for details

HCUT^4_7

dimension	21
group	Sym(7)
extreme rays	140 in two orbits
facets	474390 in 153 orbits

see files List153_4P7 for details

HMET^4_7

dimension	21
group	Sym(7)
extreme rays	3692 in 8 orbits
facets	63 in two orbits

see files 4HMET7.tex and 4HMET7.ps for details

HCUT^5_8
(the description is incomplete)

dimension	28
group	Sym(8)
extreme rays	266 in two orbits
facets	at least 408276708 in at least 11185 orbits

HMET^5_8

dimension	21
group	Sym(8)
extreme rays	55898 in 13 orbits
facets	84 in two orbits

see files List13_5HMET8 for details

HCUT^2_6

dimension	20
group	Sym(6)
extreme rays	90 in three orbits
facets	2095154 in 3086 orbits

see files List3086_2P6 for details

HMET^2_6

dimension	20
group	Sym(6)
extreme rays	12492 in 41 orbits
facets	80 in two orbits

see files 2HMET6.tex and 2HMET6.ps for details

HMET^3_7

dimension	35
group	Sym(7)
extreme rays	at least 373014230 in at least 74878 orbits
facets	140 in two orbits

HMET^2_7

dimension	35
group	Sym(7)
extreme rays	at least 243895508 in at least 49715 orbits
facets	175 in two orbits