论文标题
在平滑曲线的填料的残差集的Hausdorff尺寸上
On the Hausdorff dimension of the residual set of a packing by smooth curves
论文作者
论文摘要
让平面残留集成为通过从平面中的一个开放集中删除许多不相交的拓扑磁盘来获得的集合。我们证明，平面堆积的残差集曲线满足某种较低的曲率结合的曲线具有hausdorff尺寸的界限，但仅取决于曲率结合。作为推论，任何圆形堆积的残差集都使豪斯多夫的尺寸均匀地脱离了1。该结果概括了拉尔曼的结果，拉尔曼获得了正方形内的圆形包装的相同结论。我们还表明，如果没有较低的曲率界限，我们的定理是最佳的，并且一般不会达到。特别是，我们通过严格的凸构构建包装，其残留曲线具有尺寸1。另一方面，我们证明，严格凸出曲线的任何包装都不能具有$σ$ -finite hausdorff 1量。
Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $σ$-finite Hausdorff 1-measure.
