While research on the geometry of planar graphs has been active in the past decades, many properties of planar metrics remain mysterious. This paper studies a fundamental aspect of the planar graph geometry: covering planar metrics by a small collection of simpler metrics. Specifically, a \emph{tree cover} of a metric space $(X, \delta)$ is a collection of trees, so that every pair of points $u$ and $v$ in $X$ has a low-distortion path in at least one of the trees. The celebrated ``Dumbbell Theorem'' [ADMSS95] states that any low-dimensional Euclidean space admits a tree cover with $O(1)$ trees and distortion $1+\varepsilon$, for any fixed $\varepsilon \in (0,1)$. This result has found numerous algorithmic applications, and has been generalized to the wider family of doubling metrics [BFN19]. Does the same result hold for planar metrics? A positive answer would add another evidence to the well-observed connection between Euclidean/doubling metrics and planar metrics. In this work, we answer this fundamental question affirmatively. Specifically, we show that for any given fixed $\varepsilon \in (0,1)$, any planar metric can be covered by $O(1)$ trees with distortion $1+\varepsilon$. Our result for planar metrics follows from a rather general framework: First we reduce the problem to constructing tree covers with \emph{additive distortion}. Then we introduce the notion of \emph{shortcut partition}, and draw connection between shortcut partition and additive tree cover. Finally we prove the existence of shortcut partition for any planar metric, using new insights regarding the grid-like structure of planar graphs. [...]

翻译：尽管过去几十年中平面图几何性质的研究一直活跃，但平面度量的许多属性仍属未知。本文研究平面图几何的一个基本方面：用少量简单度量的集合来覆盖平面度量。具体而言，度量空间 $(X, \delta)$ 的**树覆盖**是一组树的集合，使得 $X$ 中任意一对点 $u$ 和 $v$ 至少在某一棵树中存在一条低失真路径。著名的"Dumbbell定理" [ADMSS95] 指出，任意低维欧氏空间均可被 $O(1)$ 棵树覆盖，且对任意固定的 $\varepsilon \in (0,1)$ 失真度为 $1+\varepsilon$。该结果已催生众多算法应用，并被推广至更广泛的加倍度量族 [BFN19]。平面度量是否具有相同性质？若答案为肯定，将为欧氏/加倍度量与平面度量之间业已观察到的紧密关联增添又一证据。在本工作中，我们对该根本问题给出肯定回答。具体而言，我们证明对任意固定的 $\varepsilon \in (0,1)$，任意平面度量均可被 $O(1)$ 棵树以失真度 $1+\varepsilon$ 覆盖。我们关于平面度量的结果源自一个相当通用的框架：首先将问题简化为构造具有**加性失真**的树覆盖，接着引入**捷径划分**概念，并建立捷径划分与加性树覆盖之间的联系，最终利用关于平面图类网格结构的新见解，证明任意平面度量均存在捷径划分。[...]