  We study the impact on mechanisms for facility location of moving from one
dimension to two (or more) dimensions and Euclidean or Manhattan distances. We
consider three fundamental axiomatic properties: anonymity which is a basic
fairness property, Pareto optimality which is one of the most important
efficiency properties, and strategy proofness which ensures agents do not have
an incentive to mis-report. We also consider how well such mechanisms can
approximate the optimal welfare. Our results are somewhat negative. Moving from
one dimension to two (or more) dimensions often makes these axiomatic
properties more difficult to achieve. For example, with two facilities in
Euclidean space or with just a single facility in Manhattan space, no mechanism
is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the
line exist with all three properties.We also show that approximation ratios may
increase when moving to two (or more) dimensions. All our impossibility results
are minimal. If we drop one of the three axioms (anonymity, Pareto optimality
or strategy proofness) multiple mechanisms satisfy the other two axioms.
