Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of oriented cuts

TL;DR

This paper demonstrates that certain cones of weighted quasi-metrics, hypermetrics, and oriented cuts can be obtained as projections of classical metric, hypermetric, and cut cones, preserving face structures.

Contribution

It establishes a geometric relationship showing these cones are projections of well-known cones, enabling face structure transfer.

Findings

Weighted quasi-metric cone is a projection of the metric cone.

Weighted quasi-hypermetric cone is a projection of the hypermetric cone.

Oriented cut cone is a projection of the cut cone.

Abstract

We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone Hypn+1 and of the cut cone Cut_{n+1}, respectively. This projection is such that if one knows all faces of an original cone then one knows all faces of the projected cone.