  We consider multiwinner elections in Euclidean space using the minimax
Chamberlin-Courant rule. In this setting, voters and candidates are embedded in
a $d$-dimensional Euclidean space, and the goal is to choose a committee of $k$
candidates so that the rank of any voter's most preferred candidate in the
committee is minimized. (The problem is also equivalent to the ordinal version
of the classical $k$-center problem.) We show that the problem is NP-hard in
any dimension $d \geq 2$, and also provably hard to approximate. Our main
results are three polynomial-time approximation schemes, each of which finds a
committee with provably good minimax score. In all cases, we show that our
approximation bounds are tight or close to tight. We mainly focus on the
$1$-Borda rule but some of our results also hold for the more general
$r$-Borda.
