### Abstract

We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in

(mathbb {R}^2)

. We show that for the *p*–*norm* (

(p ge 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 odd number 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 (

(pge 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 equals the lower bound

(2^{1-frac{1}{p}})

(as is the case for

(p=2)

and

(p=infty)

) for any

(pge 2)

.