AI 中文总结
本文针对全等欧氏球的覆盖问题，通过纯组合论证和加权硬核模型方法，改进了经典维塔利覆盖引理给出的体积下界，获得了更优的数学结果。
AI 中文摘要
设$B^d$为$\mathbb{R}^d$中的欧氏单位球，$f(B^d)$表示最大常数$c$，使得每一组有限个全等欧氏球都包含一个两两不交的子集合，其总体积至少为原集合体积的$c$倍。经典的维塔利覆盖引理给出$f(B^d)\geq3^{-d}$。本文建立了两项改进：其一，通过纯组合论证，证明对所有整数$d\geq1$，有$f(B^d)\geq \frac{2}{3^d + 2^d}$，当$d$趋于无穷时，该值相比维塔利 bound 提升了约2倍；其二，结合加权硬核模型与欧氏球相交的加权几何估计，证明对所有足够大的$d$，有$f(B^d)\geq \left( \log\frac{3}{1+\sqrt{3}} -O\left(\frac{\log d}{d}\right) \right)d\\,3^{-d}$，由此经典下界得到了阶为$d$的因子提升。
英文摘要
Let $K$ be a symmetric convex body in $\mathbb{R}^d$ and let $f(K)$ denote the largest constant $c$ such that every finite collection of translates of $K$ contains a pairwise disjoint subcollection whose total volume is at least $c$ times the volume of the union of the original collection. The classical Vitali covering lemma gives $f(K)\geq3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$ f(K)\geq \frac{2}{3^d + 2^d} $$ for every symmetric convex body $K$. This improves the Vitali bound by a factor tending to $2$ as $d$ tends to infinity. Second, using a weighted hard-core model together with a weighted geometric estimate for intersections of Euclidean balls, we show that, for all sufficiently large $d$, $$ f(B^d)\geq \left( \log\frac{3}{1+\sqrt3} -O\left(\frac{\log d}{d}\right) \right)d\,3^{-d}, $$ where $B^d$ is the unit Euclidean ball in $\mathbb{R}^d$. Thus, the classical lower bound is improved by a factor of order $d$.
Comments17 pages; any comments are welcome
