Inclusion properties of the triangular ratio metric balls

TL;DR

This paper investigates the inclusion relationships among metric balls in the unit ball domain for various metrics, providing sharp results, a conjecture, and an algorithm for visualizing triangular ratio metric balls.

Contribution

It offers new inclusion properties, proves sharp bounds, proposes a conjecture on metric ball relations, and develops an algorithm for visualizing triangular ratio metric balls.

Findings

Proved sharp inclusion results for metric balls.

Formulated a conjecture relating triangular ratio and hyperbolic balls.

Developed an algorithm for drawing metric balls and spheres.

Abstract

Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the $j^*$-metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres.