Improved Bounds for Discrete Voronoi Games

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.