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 , the length of along a balanced folding path is not larger than the maximum of its lengths at the endpoints. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counter examples.
Cite this article
Yulan Qing, Kasra Rafi, Convexity of balls in outer space. Groups Geom. Dyn. 15 (2021), no. 3, pp. 893–934DOI 10.4171/GGD/615