m-Hemi-metrics - Mathieu Dutour Sikirić

m-Hemi-metric and m-Hemi-cut Cones

m-Hemi-metric: A function d defined over (m+1)-tuples of {1,...,n} that is invariant under permutation, equals 0 if any two elements are identical, is non-negative, and satisfies the m-hemi-metric triangle inequality.
m-Hemi-metric Cone (HMET^m_n): The polyhedral cone of all m-hemi-metrics for fixed m and n.
m-Partition Hemi-metric: For an unordered (m+1)-partition of {1,...,n}, d = 0 if any two elements belong to the same subset, and 1 otherwise.
m-Hemi-cut Cone (HCUT^m_n): The positive span of all m-partition hemi-metrics.

Cone	Dim	Group	Extreme Rays	Facets	Details
HCUT²₅	10	Sym(5)	25 (2 orbits)	120 (4 orbits)	TeX \| PS
HMET²₅	10	Sym(5)	37 (3 orbits)	30 (2 orbits)	TeX \| PS
HCUT³₆	15	Sym(6)	65 (2 orbits)	4,065 (16 orbits)	TeX \| PS
HMET³₆	15	Sym(6)	287 (5 orbits)	45 (2 orbits)	TeX \| PS
HCUT²₆	20	Sym(6)	90 (3 orbits)	2,095,154 (3086 orbits)	List
HMET²₆	20	Sym(6)	12,492 (41 orbits)	80 (2 orbits)	TeX \| PS

Cone	Dim	Group	Extreme Rays	Facets	Details
HCUT⁴₇	21	Sym(7)	140 (2 orbits)	474,390 (153 orbits)	List
HMET⁴₇	21	Sym(7)	3,692 (8 orbits)	63 (2 orbits)	TeX \| PS
HCUT⁵₈	28	Sym(8)	266 (2 orbits)	≥ 408,276,708	Incomplete
HMET⁵₈	21	Sym(8)	55,898 (13 orbits)	84 (2 orbits)	List

HMET³₇: Dim 35 | ≥ 373,014,230 extreme rays (≥ 74,878 orbits) | 140 facets (2 orbits).
HMET²₇: Dim 35 | ≥ 243,895,508 extreme rays (≥ 49,715 orbits) | 175 facets (2 orbits).