关于Gromov重覆盖点的选取方法
On Gromov's Method of Selecting Heavily Covered Points
摘要：
Boros和Füredi（$d=2$）和Bárány（任意$d$）的一个结果断言，对于每个$d$，存在$c_d>0$，使得对于每个$n$ -点集$P\subset \R^d$，$\R^d$的某个点被至少$c_d{n\choose d+1}$的$d$ -单形覆盖，这些单形由$P$的点所张成。$c_d$的最大可能价值一直是正在进行的研究的主题。最近Gromov通过引入一种新的拓扑证明方法大大改进了现有的下界。我们提供了一个组合的格罗莫夫的方法组成部分的阐述，在combinatorialists和离散geometers访问的条款，我们调查他的方法的限制。特别是，我们给出了更严格的界限上的\emph{cofilling profiles}的$(n-1)$ -单纯形。这些界限产生了一个小的改进格罗莫夫的下限$c_d$为大$d$，但他们也表明，通过{ \cofilling }配置文件单独的进一步改进的空间是相当小的。我们还证明了一个稍微好一点的下限$c_3$的方法，使用一个额外的结构，除了{ \cofilling }配置文件。我们制定了一个组合极值问题，其解决方案可能会导致$c_d$的一个严格的下限。
英文摘要：
A result of Boros and Füredi ($d=2$) and of Bárány (arbitrary $d$) asserts that for every $d$ there exists $c_d>0$ such that for every $n$-point set $P\subset \R^d$, some point of $\R^d$ is covered by at least $c_d{n\choose d+1}$ of the $d$-simplices spanned by the points of $P$. The largest possible value of $c_d$ has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov's approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the \emph{cofilling profiles} for the $(n-1)$-simplex. These bounds yield a minor improvement over Gromov's lower bounds on $c_d$ for large $d$, but they also show that the room for further improvement through the {\cofilling} profiles alone is quite small. We also prove a slightly better lower bound for $c_3$ by an approach using an additional structure besides the {\cofilling} profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for $c_d$.
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