A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture

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…

关于小维度下哈德维格覆盖问题的研究

On Hadwiger's covering problem in small dimensions

本文针对小维度凸体覆盖问题，利用 John 椭球平移、格覆盖及 quermassintegral 估计，给出 $H_5$ 至 $H_8$ 及对称情形 $H_4^s$ 至 $H_6^s$ 的新上界，并讨论改进一般渐近界的方法。

Functionals on the Spaces of n-Dimensional Convex Bodies

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on…

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.

典型范数的Hadwiger--Nelson问题

Hadwiger--Nelson Problem for Typical Norms

本文研究典型范数的Hadwiger--Nelson问题，将单位距离图色数的上界从指数改进为线性（至多2d且紧），并借助高维矩阵推广的Lonely runner猜想解决了相关猜想。

The covering radius and a discrete surface area for non-hollow simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino \& Schymura (2017) that the $d$-th covering minimum of the standard terminal $n$-simplex equals $d/2$, for every $n>d$. We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the cover…