Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any $(α,β)$-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in $(α,β)$-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on $(α,β)$-covered polygons, including an algorithm that computes the closest pair of a set of $m$ points in an $(α,β)$-covered polygon with $n$ vertices that runs in $O(n + m\log{n})$ expected time.

翻译：许多算法问题在具有有界倍率维度的度量空间中可以像在欧几里得空间中一样高效（或近乎高效）地求解。然而，由简单多边形内点集构成的、装备测地距离的度量空间未必具有有界倍率维度。为此，我们针对两种经典胖度定义研究胖多边形的倍率维度。我们证明：局部胖简单多边形不一定具有有界倍率维度，而任意 $(α,β)$-覆盖多边形（即使包含孔洞）均具有有界倍率维度。进一步，我们研究 $(α,β)$-覆盖多边形（可能含孔洞）中测地凸集的周长，并证明该周长至多为集合欧几里得直径的常数倍。基于这两项结果，我们获得了 $(α,β)$-覆盖多边形上若干问题的新结论，包括一种在具有 $n$ 个顶点的 $(α,β)$-覆盖多边形中计算 $m$ 个点最近点对的算法，其期望运行时间为 $O(n + m\log{n})$。