Intrinsic quasi-metrics

TL;DR

This paper introduces and analyzes intrinsic quasi-metrics in convex domains, establishing their properties and connections to existing metrics, with implications for geometric analysis.

Contribution

The paper defines a new intrinsic quasi-metric $w_G$ for convex domains and studies its properties and relation to the point pair function $p_G$ and the triangular ratio metric.

Findings

Both $p_G$ and $w_G$ are quasi-metrics with sharp properties.

Connections between $w_G$, $p_G$, and the triangular ratio metric are established.

New inequalities and bounds for these metrics are derived.

Abstract

The point pair function $p_G$ defined in a domain $G\subsetneq\mathbb{R}^n$ is shown to be a quasi-metric and its other properties are studied. For a convex domain $G\subsetneq\mathbb{R}^n$, a new intrinsic quasi-metric called the function $w_G$ is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.