论文标题
在飞机上耐受性的Tverberg分区的新下限
New Lower Bounds for Tverberg Partitions with Tolerance in the Plane
论文作者
论文摘要
让$ p $是$ d $维空间中的$ n $点。 Tverberg的定理说，如果$ n $至少为$（k-1）（d+1）+1 $，则可以将$ p $划分为$ k $ sets，其凸壳相交。使用此属性的分区称为{\ em tverberg分区}。如果该分区从$ p $中取出任何一组$ t $点后，分区仍然是tverberg分区，则分区具有公差$ t $。如果$ n $足够大，则在任何维度上都存在耐受性的Tverberg分区。令$ n（d，k，t）$为$ n $的最小值，以便在$ \ mathbb {r}^d $中的任何一组$ n $点都存在容忍的tverberg分区。 $ n（d，k，t）$的精确值很少。 在本文中，我们为$ n（2,2,2）$建立了一个新的紧身限制。我们还证明了$ n（2，k，t）$ for $ k \ ge 2 $和$ t \ ge 1 $上的许多新的下限。
Let $P$ be a set $n$ points in a $d$-dimensional space. Tverberg's theorem says that, if $n$ is at least $(k-1)(d+1)+1$, then $P$ can be partitioned into $k$ sets whose convex hulls intersect. Partitions with this property are called {\em Tverberg partitions}. A partition has tolerance $t$ if the partition remains a Tverberg partition after removal of any set of $t$ points from $P$. Tolerant Tverberg partitions exist in any dimension provided that $n$ is sufficiently large. Let $N(d,k,t)$ be the smallest value of $n$ such that tolerant Tverberg partitions exist for any set of $n$ points in $\mathbb{R}^d$. Only few exact values of $N(d,k,t)$ are known. In this paper we establish a new tight bound for $N(2,2,2)$. We also prove many new lower bounds on $N(2,k,t)$ for $k\ge 2$ and $t\ge 1$.
