Close-to-convexity of quasihyperbolic and $j$-metric balls

TL;DR

This paper investigates the close-to-convexity properties of metric balls defined by the quasihyperbolic and j-metrics, showing that small-radius balls are close-to-convex in certain domains.

Contribution

It establishes that small-radius j-metric balls are close-to-convex in general subdomains, and quasihyperbolic balls are close-to-convex in punctured space.

Findings

Small-radius j-metric balls are close-to-convex in subdomains.

Small-radius quasihyperbolic balls are close-to-convex in punctured space.

Provides conditions for close-to-convexity of metric balls.

Abstract

We will consider close-to-convexity of the metric balls defined by the quasihyperbolic metric and the $j$-metric. We will show that the $j$-metric balls with small radii are close-to-convex in general subdomains of $\Rn$ and the quasihyperbolic balls with small radii are close-to-convex in the punctured space.