In the planar one-round discrete Voronoi game, two players $\mathcal{P}$ and $\mathcal{Q}$ compete over a set $V$ of $n$ voters represented by points in $\mathbb{R}^2$. First, $\mathcal{P}$ places a set $P$ of $k$ points, then $\mathcal{Q}$ places a set $Q$ of $\ell$ points, and then each voter $v\in V$ is won by the player who has placed a point closest to $v$. It is well known that if $k=\ell=1$, then $\mathcal{P}$ can always win $n/3$ voters and that this is worst-case optimal. We study the setting where $k>1$ and $\ell=1$. We present lower bounds on the number of voters that $\mathcal{P}$ can always win, which improve the existing bounds for all $k\geq 4$. As a by-product, we obtain improved bounds on small $\varepsilon$-nets for convex ranges. These results are for the $L_2$ metric. We also obtain lower bounds on the number of voters that $\mathcal{P}$ can always win when distances are measured in the $L_1$ metric.

翻译：在平面单轮离散Voronoi博弈中，两位参与者$\mathcal{P}$和$\mathcal{Q}$在由$\mathbb{R}^2$中点表示的$n$个投票者集合$V$上进行竞争。首先，$\mathcal{P}$放置包含$k$个点的集合$P$，随后$\mathcal{Q}$放置包含$\ell$个点的集合$Q$，接着每个投票者$v\in V$由放置点距离$v$最近的参与者赢得。众所周知，当$k=\ell=1$时，$\mathcal{P}$总能赢得$n/3$个投票者，且该结果在最坏情况下是最优的。我们研究$k>1$且$\ell=1$的情形。本文提出了$\mathcal{P}$总能赢得的投票者数量的下界，该结果改进了现有对所有$k\geq 4$情形的界。作为副产品，我们获得了凸区域小$\varepsilon$-网的改进界。以上结果均针对$L_2$度量。我们还得到了在$L_1$度量下$\mathcal{P}$总能赢得的投票者数量的下界。