关于平面集合的 $\varepsilon$-邻域边界：奇点、全局结构与曲率
On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets: Singularities, Global Structure, and Curvature
AI总结：
本文研究紧致平面集合 $\varepsilon$-邻域边界的几何与拓扑性质，开发新技术将奇点分为八类，刻画了补集正可达的局部几何条件，给出边界一致可求长的充分条件，并证明曲率在边界 Jordan 曲线子集上几乎处处有定义。
摘要：
我们研究紧致平面集合 $E \subset \mathbb{R}^2$ 的闭 $\varepsilon$-邻域 $E_\varepsilon = \{x \in \mathbb{R}^2 \\, : \\, \textrm{dist}(x, E) \leq \varepsilon \}$ 边界的几何性质、拓扑性质与平滑性。我们开发了一种分析边界的新技术，并借此将边界 $\partial E_\varepsilon$ 上的奇点（即非光滑点）划分为八个类别。我们证明，奇点集要么是可数的，要么是一个可数集与一个闭的、全不连通且无处稠密集的不相交并集。此外，我们用局部几何刻画了那些补集 $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ 为正可达集的 $\varepsilon$-邻域。已知对于所有有界集合 $E \subset \mathbb{R}^d$ 和所有 $\varepsilon > 0$，边界 $\partial E_\varepsilon$ 是 $(d-1)$-可求长的。在此基础上，我们给出了边界一致可求长的充分条件，并给出了一个非 Ahlfors 正则的平面 $\varepsilon$-邻域示例。在拓扑结构方面，我们证明对于紧致集合 $E$ 和 $\varepsilon > 0$，边界 $\partial E_\varepsilon$ 可以表示为至多可数无穷多条 Jordan 曲线的不相交并集，以及一个可能不可数的、全不连通的奇点集。最后，我们证明曲率在边界的 Jordan 曲线子集上几乎处处有定义。
英文摘要：
We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of compact planar sets $E \subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary, and use this to obtain a classification of singularities (i.e.~non-smooth points) on $\partial E_\varepsilon$ into eight categories. We show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set. Furthermore, we characterise, in terms of local geometry, those $\varepsilon$-neighbourhoods whose complement $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ is a set with positive reach. It is known that for all bounded $E \subset \mathbb{R}^d$ and all $\varepsilon > 0$, the boundary $\partial E_\varepsilon$ is $(d-1)$-rectifiable. Improving on this, we identify a sufficient condition for the boundary to be uniformly rectifiable, and provide an example of a planar $\varepsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, we show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.
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