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：许多算法问题可以在有界加倍维数的度量空间中，以与欧氏空间几乎相同的效率得到解决。遗憾的是，由简单多边形内点集构成的测地距离度量空间并不一定具有有界加倍维数。因此，我们针对两种常见的胖度定义，研究了胖多边形的加倍维数。我们证明了局部胖简单多边形并不总是具有有界加倍维数，而任何$(α,β)$-覆盖多边形（即使带孔洞）都具有有界加倍维数。我们还研究了$(α,β)$-覆盖多边形（可能带孔洞）中测地凸集的周长，并证明了该周长至多与集合的欧氏直径成正比。利用这两个结论，我们获得了关于$(α,β)$-覆盖多边形若干问题的新结果，包括一个在具有$n$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$O(n + m\log{n})$。