The isometries of the cut, metric and hypermetric cones

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cutn and the metric cone Metn both consist of the isometries induced by the permutations on {1, . . . , n}; that is, Is(Cutn) = Is(Metn) ≃ Sym(n) for n ≥ 5. For n = 4 we have Is(Cut4) = Is(Met4) ≃ Sym(3) × Sym(4). This result can be extended to cones...

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Bibliographic Details
Main Authors: Deza, Antoine., Goldengorin, Boris., Pasechnik, Dmitrii V.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2011
Subjects:
DRNTU::Science::Mathematics::Geometry
Online Access:https://hdl.handle.net/10356/92362
http://hdl.handle.net/10220/6867
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Institution: Nanyang Technological University
Language: English
Description
Summary:We show that the symmetry groups of the cut cone Cutn and the metric cone Metn both consist of the isometries induced by the permutations on {1, . . . , n}; that is, Is(Cutn) = Is(Metn) ≃ Sym(n) for n ≥ 5. For n = 4 we have Is(Cut4) = Is(Met4) ≃ Sym(3) × Sym(4). This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hypn) ≃ Sym(n) for n ≥ 5, where Hypn denotes the hypermetric cone.

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