Variance estimates for random disc-polygons in smooth convex discs

Separating circles on the sphere by polygonal tilings

We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous sta…

关于具有更高平滑度的平面凸体中的广义圆盘多边形

On generalized disc-polygons in plane convex bodies with a higher degree of smoothness

本文证明了平滑平面凸体中随机 $L$-凸多边形的顶点数期望与遗漏面积的幂级数展开式，扩展了已有渐近公式并推及 $L$-凸浮体与相对仿射表面积。

Area of Polygons in Hyperbolic Geometry

Consider the Poincare disc model for hyperbolic geometry. In this paper, a convenient computational formula is developed along with an aesthetic geometric interpretation. Two proofs, one geometric and one analytical, of each result are given.

欧氏平面中多边形的一些极值问题

Some extremal problems for polygons in the Euclidean plane

本文研究了欧氏平面中凸多边形及其周长相关的极值问题，提出了若干表述简单但证明复杂的结果，并讨论了未解决的问题。

Classification and enumeration of lattice polygons in a disc

In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of given area. Since then, this problem and its analogues have been studied by many authors, including $\mathrm{B\acute{a}r\acute{a}ny}$, Lagarias, Pach, Santos, Ziegler and Zong. Recently, Zong proposed two computer programs to prove Hadwiger's covering conjecture and Borsuk's partition problem, respectively, based on enumeration of the convex lattice polytopes contained in certain balls. For this purpose, similar to $\mathrm{B\acute{a}r\acute{a}ny}$ and Pach's work on volume and Liu and Zong's work on cardin…

New Properties and Invariants of Harmonic Polygons

Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.

单位圆盘中多边形上的几何

Geometries on Polygons in the unit disc

本文在单位圆盘内的凸多边形上定义了一种新度量，使其测地线恰为给定曲线族与多边形的交集，并证明在直线情形下不等距于希尔伯特度量的凸域，且推广至更一般的唯一测地空间。

正多边形的单面铺嵌

On monohedral tilings of a regular polygon

本文连接了正方形和圆盘的三块铺嵌结果，刻画了任意正n边形（n≥5）被至多三个拓扑圆盘单面铺嵌的情况。

Intersecting the sides of a polygon

Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the dynamics of the map $S$. Namely, we address the question of whether a convex polygon stays convex under iterations of $S$. Computer experiments suggest that this almost never happens. We prove that indeed the set of polygons which remain convex under iterations of $S$ has measure zero, and moreover it is an algebraic subvariety of codimension two. We also dis…

格多边形的体积多项式

The volume polynomial of lattice polygons

本文证明每个具有非负整数系数的不定二次型均为一对格多边形的体积多项式，从而解决了平面上两个体的 Heine-Shephard 离散问题，并应用于构造具有给定相交数和自相交数的平面 tropical 曲线或环面曲面除子对。