  We extend the recently introduced framework of metric distortion to
multiwinner voting. In this framework, $n$ agents and $m$ alternatives are
located in an underlying metric space. The exact distances between agents and
alternatives are unknown. Instead, each agent provides a ranking of the
alternatives, ordered from the closest to the farthest. Typically, the goal is
to select a single alternative that approximately minimizes the total distance
from the agents, and the worst-case approximation ratio is termed distortion.
In the case of multiwinner voting, the goal is to select a committee of $k$
alternatives that (approximately) minimizes the total cost to all agents. We
consider the scenario where the cost of an agent for a committee is her
distance from the $q$-th closest alternative in the committee. We reveal a
surprising trichotomy on the distortion of multiwinner voting rules in terms of
$k$ and $q$: The distortion is unbounded when $q \leq k/3$, asymptotically
linear in the number of agents when $k/3 < q \leq k/2$, and constant when $q >
k/2$.
