Convexity of balls in the outer space

TL;DR

This paper investigates the convexity properties of geodesics and balls in Outer space with the Lipschitz metric, introducing balanced folding paths and demonstrating weak convexity of out-going balls with sharp counterexamples.

Contribution

It introduces balanced folding paths in Outer space and proves weak convexity of out-going balls, providing counterexamples to show the results are sharp.

Findings

Out-going balls are weakly convex in Outer space.

Balanced folding paths preserve length bounds for loops.

Counterexamples demonstrate the sharpness of the convexity results.

Abstract

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $\alpha$, the length of $\alpha$ along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.