发表机构
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明可将平面凸集切割为等面积且保持胖性的部分，并基于此构造图，使图距离与欧几里得距离之差满足给定界，解决了 Benjamini 等人问题的弱形式。
AI 中文摘要
能否在平面上的整数点集 ${\mathbb Z}^2$ 上构造一个图 $G$，使得 $G$ 中任意两个顶点之间的最短路径长度与它们的欧几里得距离之差至多为一个绝对常数？这个问题由 Benjamini、Erd\H os、Kleiner、Kozma、Schramm 以及第一作者提出，长期悬而未决。我们对该问题的一个较弱形式给出了肯定回答，其基础是以下具有独立意义的几何命题：存在常数 $c>0$，使得对于每个 $i=1,2,\ldots,$ 每个 $\rho$-胖平面凸集 $S$ 都可以被切割成 $2^i$ 个面积相等的凸片，每个凸片至少是 $c\rho$-胖的。（若一个凸集的内切圆半径与外接圆半径之比至少为 $\rho$，则称其为 $\rho$-胖的。）我们证明，在 ${\mathbb Z}^2$ 的一个放大副本上存在一个（无权的）生成子图 $G$，使得对于任意一对欧几里得距离为 $d$ 的顶点，它们在 $G$ 中的最短路径距离介于 $d-O(1)$ 与 $d+o(d^{5/6})$ 之间。同样的界可以通过一个顶点集为 ${\mathbb Z}^2$ 的平面图实现，其中每条边连接欧几里得距离至多为 2 的两个顶点。
英文摘要
Can one construct a graph $G$ on the set of integer points ${\mathbb Z}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\ldots,$ every $ρ$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $cρ$-fat. (A convex set is $ρ$-fat if the ratio of its inradius to its circumradius is at least $ρ$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\mathbb Z}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5/6})$. The same bound can be achieved by a planar graph with vertex set ${\mathbb Z}^2$, in which every edge joins two vertices at Euclidean distance at most 2.
Comments25 pages, 9 figures
