  We initiate the study of the capacity constrained facility location problem
from a mechanism design perspective. The capacity constrained setting leads to
a new strategic environment where a facility serves a subset of the population,
which is endogenously determined by the ex-post Nash equilibrium of an induced
subgame and is not directly controlled by the mechanism designer. Our focus is
on mechanisms that are ex-post dominant-strategy incentive compatible (DIC) at
the reporting stage. We provide a complete characterization of DIC mechanisms
via the family of Generalized Median Mechanisms (GMMs). In general, the social
welfare optimal mechanism is not DIC. Adopting the worst-case approximation
measure, we attain tight lower bounds on the approximation ratio of any DIC
mechanism. The well-known median mechanism is shown to be optimal among the
family of DIC mechanisms for certain capacity ranges. Surprisingly, the
framework we introduce provides a new characterization for the family of GMMs,
and is responsive to gaps in the current social choice literature highlighted
by Border and Jordan (1983) and Barbar{\`a}, Mass{\'o} and Serizawa (1998).
