The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace. This Frechet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Frechet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.

中文翻译：

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Billera-Holmes-Vogtmann 树空间中 Fréchet 均值的属性

加权树的 Billera-Holmes-Vogtmann (BHV) 空间可以嵌入到欧几里得空间中，但外在欧几里得均值通常位于树空间之外。Sturm 表明内在 Frechet 均值存在并且在树空间中是唯一的。这个 Frechet 均值可以用迭代算法近似，但算法收敛的界限是未知的，并且没有其他已知的多项式算法来计算 Frechet 均值，甚至没有出现在均值中的边。我们给出边缘在 Frechet 均值中的第一个充分必要条件。条件是边权重不等式的形式。这些条件为查找包含 Frechet 均值的树空间 orthant 提供了一个预处理步骤。这项工作推广到orthant空间。