用交叉多胞体覆盖交叉多胞体

Covering the crosspolytope with its smaller homothetic copies

In 1957, Hadwiger made the famous conjecture that any convex body of $n$-dimensional Euclidean space $\mathbb{E}^n$ can be covered by $2^n$ smaller positive homothetic copies. Up to now, this conjecture is still open for all $n\geq 3$. Denote by $γ_{m}(K)$ the smallest positive number $λ$ such that $K$ can be covered by $m$ translations of $λK$. The values of $γ_m(K)$ for some particular $m$ and $K$ have been studied. In this article, we will focus on the situation where $K$ is the unit crosspolytope of the three-dimensional.

齐性空间的度量熵

Metric Entropy of Homogeneous Spaces

对于度量空间的（紧）子集$K$和$\varepsilon > 0$，{ \em覆盖数} $N(K , \varepsilon )$定义为半径为$\varepsilon$且其并集覆盖$K$的球的最小数目。{ \em度量熵}的知识，即，度量空间（族）的覆盖数的渐近行为在数学的许多领域（几何、泛函分析、概率、编码理论等）中是重要的。在本文中，我们给出了关于一些自然度量，最值得注意的是由算子范数诱导的一类酉（或正交）群的齐性空间的覆盖数的渐近正确估计。这推广了早先作者关于格拉斯曼流形的覆盖数的结果;推广的动机是应用到非交换概率和算子代数。在这个过程中，我们给出了一类非黎曼度量结构的$U(n)$（或$SO(m)$）中测地线的一个特征。

森林中最小直径的封闭逃逸路径：勒贝格通用覆盖问题的对偶表述

Closed escape path of smallest diameter in forest: dual formulation of Lebesgue's universal covering problem

本文通过森林逃逸问题的最小直径对偶，将勒贝格通用覆盖问题转化为优化问题，并给出离散化与严格认证的数值方法。

Homothetic covering of convex hulls of compact convex sets

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $λ\in[0,1]$ such that $K$ can be covered by $m$ translates of $λK$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.

Improved bounds for Hadwiger's covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal co…