Ákos G. Horváth
This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness.
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Given a compact connected set $E$ in the unit disk $\mathbb{B}^{2}$, we give a new upper bound for the conformal capacity of the condenser $(\mathbb{B}^{2}, E)$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$, we construct a set of hyperbolic diameter $t$ and apply novel numerical methods to show that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is called a Reuleaux triangle in hyperbolic geometry and it has constant hyperbolic width equal to $t$.
On the perimeter, diameter and circumradius of ordinary hyperbolic reduced polygons
本文研究双曲平面中的普通约化多边形，回答了 Lassak 关于其周长、直径和外接圆半径的问题，并将 Fabińska 的相关结果推广到双曲情形。
Lipschitzness of the width and diameter functions of convex bodies in $\mathbb R^n$
研究了凸体在R^n中宽度和直径函数的Lipschitz性，通过直径和厚度确定其Lipschitz常数，并提出厚度的对偶方法。
We show that expander graphs must have Gromov-hyperbolicity at least proportional to their diameter, with a constant of proportionality depending only on the expansion constant and maximal degree. In other words, expanders contain geodesic triangles which are $Ω(\mathop{\rm diam} Γ)$-thick.
The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $Δ(L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $Δ(C)$ of a convex body $C \subset S^2$ we mean the minimal value of $Δ(L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $Δ(Z) < Δ(R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $Δ(R) < \fracπ{2}$, in terms of its thickness.
Let $O$ be a closed $n$-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of $O$ is bounded above by $$\frac{c_1\log vol(O) + c_2}{h(O)},$$ where $h(O)$ is the Cheeger constant of $O$, $vol(O)$ is its volume, and constants $c_1$, $c_2$ depend only on $n$.
We show that closed arithmetic hyperbolic n-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic n-orbifolds.
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies …
Area and diameter gaps for hyperbolic monotiles
本文针对双曲平面及闭曲面上的测地线单调瓦片，证明了直径与面积的正下界，并给出显式常数，同时构造了边数可任意大但直径趋于零的示例，表明边数需附加条件限制。
Reduced polygons in the hyperbolic plane
本文研究双曲平面中约化多边形，通过厚度概念刻画其极小性，并利用厚度估计其直径，揭示该类几何对象的性质。
Width of convex bodies in hyperbolic space
本文研究双曲空间中凸体的宽度、厚度、常宽体、约化体、完备体与常直径体，证明常宽体严格凸且常宽条件等价于常直径与完备性，并证明双曲正四面体不是约化体。
The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance of the centers of the two semicircles bounding it. The thickness $Δ(C)$ of $C$ is the minimum thickness of a lune containing $C$. We say that a spherical convex body $R$ is reduced if $Δ(Z) < Δ(R)$ for every spherical convex body $Z \subset R$ different from $R$. Our main theorem permits to describe the shape of reduced bodies of thickness below $\fracπ{2}$. …
Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions
该研究探讨高维双曲空间中径向测度的壳几何、金字塔极限与高斯相变，明确有效半径的作用，揭示两类高斯的临界阶差异及对应收敛性质。
Spherical quadrilateral with three right angles and its application for diameter of extreme points of a convex body
本文证明了具有三个直角的球面四边形边长关系定理，并以此证明二维球面凸体极值点集合直径的下界估计，该估计是紧的无法改进。
The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we…
We present some relationships between the diameter, width and thickness of a reduced convex body on the $d$-dimensional sphere. We apply the obtained properties to recognize if a Wulff shape in the Euclidean $d$-space is self-dual.
On reduced spherical bodies
本论文汇集五项研究，系统探讨 $S^d$ 上约化球面凸体及等宽体的形状、性质、直径与覆盖圆盘半径。
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