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 endpoints. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counter examples.

中文翻译：

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外层空间球的凸度

在本文中，我们研究了配备 Lipschitz 度量的外层空间测地线和球的凸性特性。我们引入了一类称为平衡折叠路径的测地线，并表明，对于每个循环 $\alpha$，沿着平衡折叠路径的 $\alpha$ 的长度不大于其端点处的最大长度。这意味着向外的球是弱凸的。然后我们通过提供几个反例来证明这些结果是尖锐的。