Local convexity properties of Apollonian and Seittenranta's metric balls

TL;DR

This paper investigates the local convexity characteristics of metric balls within Apollonian and Seittenranta's metrics across various geometric domains, enhancing understanding of their shape and convexity properties.

Contribution

It provides new insights into the local convexity of metric balls in Apollonian and Seittenranta's metrics in specific punctured and starlike domains.

Findings

Balls in Apollonian metric are locally convex in certain domains.

Seittenranta's metric balls exhibit convexity properties in punctured spaces.

Results improve geometric understanding of these metrics in complex domains.

Abstract

We consider local convexity properties of balls in the Apollonian and Seittenranta's metrics. Balls in the Apollonian metric are considered in the twice punctured space and starlike domains. Balls in Seittenranta's metric are considered in the twice punctured space and in the punctured ball.