**Quasi-metrics**

When I discovered hypermetric inequalities (an attempt to characterize -embeddable metrics), there turned out to be applications in the geometry of numbers. I studied the generalization to quasi-metrics, hoping for similar applications.

**1. Definition and examples **

A quasi-metric is a nonnegative function on such that

- ;
- , , , .

Example 1Minkowski gauge of a compact convex set containing the origin.

Example 2Sorgenfrei quasi-metric on is if , otherwise.

Example 3Path quasi-metric of a directed graph.

Variant: [Chartrand, Erwin, Raines, Zhang 1999] define the strong quasi-metric in a digraph.

Example 4Thurston quasi-metric between Riemannian manifolds.

** 1.1. Weightable quasi-semi-metrics **

Say is weightable if there exists function such that

A quasi-semi-metric is weightable iff its satisfies a cyclic inequality…

Any finite quasi-semi-metric is the shortest path quasi-semi-metric of a digraph. It is weighted iff the graph is na bidirectional tree.

Example 5The hitting time quasi-metric is the expected number if step of random walk needed to go from to .

It is weightable. It is proportional to the effective resistance metric.

Example 6Oriented multicut metrics.

An oriented multicut of a set is a partition with an ordering of the pieces. There is a corresponding quasi-semi-metric. It is weightable iff there are only two pieces.

** 1.2. Semantics of computation **

Say a poset , having a smallest element, is dspo if each directed subset has a supremum. A Scott domain is a dspo where all sets are directed with supremum equal to , and each consistent has a supremum in .

Example 7. All words over a finite alphabet, with prefix order. All vague real numbers (i.e. closed intervals with reverse inclusion order).

** 1.3. Cones **

All weighted quasi-semi-metrics on points form a polyhedral cone of dimension .

** 1.4. Derivation **

-derivation is Gromov product with vertex , i.e.

[Makarychev, Makarychev]: Most weightable quasi-semi-metrics arise as Gromov product of some semi-metric.

Example 8The -derivation of the -distance is called -quasi-metric and denoted by .

** 1.5. Generalization of -metrics **

Theorem 1A quasi-semi-metric embeds into iff it belongs to the cone generated by oriented cuts.

Note that an oriented cut quasi-semi-metric is the -derivation of the corresponding cut metric. It is weightable.

Example 9embeds in if .

is the square of a -quasi-semi-metric.

Theorem 2A quasi-semi-metric embeds into iff it is the quasi-semi-metric of subsets of some measure space.

What does not work: Hypercube does not have a natural quasi-semi-metric. Indeed, there are a lot of possible orientations on the hypercube. This is a whole subject in itself.

** 1.6. Hypermetric inequalities **

Hypermetric inequalities describe certain facets of the cut cone.

Given integers which sum up to , metric satisfies

Say a metric is a *hypermetric* if it satisfies all these inequalities.

Given a lattice, and a maximum ball which contains no nonzero vectors of the lattice. It may contain lattice points on its boundary. Such a set is called a constellation. The square of Euclidean metric restricted to a constellation satisfies all hypermetric inequality. Conversely, every hypermetric is isometric to some constellation [Assouad-Deza].

This has a partial generalization to quasi-metrics. As yet, it looks rather ugly.