We consider the problem of locating a facility to serve a set of agents
located along a line. The Nash welfare objective function, defined as the
product of the agents' utilities, is known to provide a compromise between
fairness and efficiency in resource allocation problems. We apply this welfare
notion to the facility location problem, converting individual costs to
utilities and analyzing the facility placement that maximizes the Nash welfare.
We give a polynomial-time approximation algorithm to compute this facility
location, and prove results suggesting that it achieves a good balance of
fairness and efficiency. Finally, we take a mechanism design perspective and
propose a strategy-proof mechanism with a bounded approximation ratio for Nash
welfare