平面集合的 $\varepsilon$-邻域边界，第二部分：全局结构与曲率

平面中具有无处光滑边界的ε-邻域

$\varepsilon$-neighbourhoods in the Plane with a Nowhere-smooth Boundary

该研究构造了两个平面集合E的例子，分别证明其ε-邻域边界无处C¹光滑（奇点可数稠密）及兼具不可数非整数Hausdorff维数且曲率未定义的子集，构造基于星形集是子集ε-邻域的特征。

凸体的内半径和 $\varepsilon$-内邻域的简单界

Simple bounds for the inradius and $\varepsilon$-inner neighbourhood of a convex body

本文证明了凸体内半径与体积/表面积相当，并给出用体积和内半径表示的与边界$\varepsilon$-邻域交集体积相当的公式，应用于具有凸孔的自射影吸引子。

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

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。

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the $(d-1)$-dimensional sphere and have the property that the images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ are codimension $1$ submanifolds contained in the interior of $K$. Moreover, as the parameter $\varepsilon$ …

Surface dimension, tiles, and synchronising automata

We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where $G_{\varepsilon}$ denotes the $\varepsilon$-neighbourhood of~$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and find them …