Local convexity of metric balls

TL;DR

This paper investigates the local convexity of metric balls in various geometric contexts, focusing on the triangular ratio and visual angle metrics within specific subdomains of real space.

Contribution

It provides new insights into the convexity and inclusion properties of metric balls in hyperbolic and related metrics in certain domains.

Findings

Convexity properties of triangular ratio metric balls are characterized.

Inclusion relations of visual angle metric balls are established.

Results are applied to hyperbolic type metric balls in specific domains.

Abstract

We study local convexity properties of the triangular ratio metric balls in proper subdomains of the real coordinate space. We also study inclusion properties of the visual angle metric balls and related hyperbolic type metric balls in the complement of the origin and the upper half space.