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We consider the facility location problem in two dimensions. In particular,we consider a setting where agents have Euclidean preferences, defined by theirideal points, for a facility to be located in $\mathbb{R}^2$. We show that forthe $p-norm$ ($p \geq 1$) objective, the coordinate-wise median mechanism (CM)has the lowest worst-case approximation ratio in the class of deterministic,anonymous, and strategyproof mechanisms. For the minisum objective and an oddnumber of agents $n$, we show that CM has a worst-case approximation ratio (AR)of $\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}$. For the $p-norm$ social cost objective($p\geq 2$), we find that the AR for CM is bounded above by$2^{\frac{3}{2}-\frac{2}{p}}$. We conjecture that the AR of CM actually equalsthe lower bound $2^{1-\frac{1}{p}}$ (as is the case for $p=2$ and $p=\infty$)for any $p\geq 2$.

Optimality of the coordinate-wise median mechanism for strategyproof facility location in two dimensions