  The concept of distance rationalizability of social choice rules has been
explored in recent years by several authors. We deal here with several
foundational questions, and unify, correct, and generalize previous work. For
example, we study a new question involving uniqueness of representation in the
distance rationalizability framework, and present a counterexample.
  For rules satisfying various axiomatic properties such as anonymity,
neutrality and homogeneity, the standard profile representation of input can be
compressed substantially. We explain in detail using quotient constructions and
symmetry groups how distance rationalizability is interpreted in this
situation. This enables us to connect the theory of distance rationalizability
with geometric concepts such as Earth Mover distance and optimal
transportation. We expect this connection to prove fruitful in future work.
  We improve on the best-known sufficient conditions for rules rationalized via
votewise distances to satisfy anonymity, neutrality, homogeneity, consistency
and continuity. This leads to a class of well-behaved rules which deserve
closer scrutiny in future.
