We consider the space of probability measures on a discrete set X , endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y ⊆ X , it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.

中文翻译：

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关于离散最优传输中的测地线几何。

我们考虑了具有动态最优传输度量的离散集X上的概率测度空间。给定子集Y⊆X中支持的两个概率测度，很自然地要问，它们是否始终可以通过恒速测地线与Y的支持联系起来。我们的主要结果肯定地回答了这个问题，在本文介绍的适当的几何条件下对Y进行了求解。该证明依赖于离散哈密顿-雅各比方程子解决方案的扩展结果，这是具有独立意义的。