Data mining for cones of metrics, quasi-metrics, hemi-metrics and super-metrics

TL;DR

This paper computationally explores complex convex cones related to metrics, quasi-metrics, hemi-metrics, and super-metrics, providing new data and criteria for their structure using adapted algorithms.

Contribution

It introduces computational methods to analyze difficult cases of convex cones of generalized metrics and provides detailed structural data and adjacency criteria.

Findings

Computed the first difficult cases of convex cones of generalized metrics.

Collected data on facets, extreme rays, and their symmetries.

Studied adjacency criteria for cone skeletons and duals.

Abstract

Using some adaptations of the adjacency decomposition method \cite{CR} and the program {\it cdd} (~\cite{Fu}), we compute the first computationally difficult cases of convex cones of $m$-ary and oriented analogs of semi-metrics and cut semi-metrics, which were introduced in \cite{DR2} and \cite{DP}. We considered also more general notion of $(m,s)$-super-metric and corresponding cones. The data on related cones - the number of facets, of extreme rays, of their orbits and diameters - are collected in Table \ref{tab:MainLovelyTable}. We study also criterion of adjacency for skeletons of those cones and their duals. Some families of extreme rays and operations on them are also given.