  The design of algorithms for political redistricting generally takes one of
two approaches: optimize an objective such as compactness or, drawing on fair
division, construct a protocol whose outcomes guarantee partisan fairness. We
aim to have the best of both worlds by optimizing an objective subject to a
binary fairness constraint. As the fairness constraint we adopt the geometric
target, which requires the number of seats won by each party to be at least the
average (rounded down) of its outcomes under the worst and best partitions of
the state.
  To study the feasibility of this approach, we introduce a new model of
redistricting that closely mirrors the classic model of cake-cutting. This
model has two innovative features. First, in any part of the state there is an
underlying 'density' of voters with political leanings toward any given party,
making it impossible to finely separate voters for different parties into
different districts. This captures a realistic constraint that previously
existing theoretical models of redistricting tend to ignore. Second, parties
may disagree on the distribution of voters - whether by genuine disagreement or
attempted strategic behavior. In the absence of a 'ground truth' distribution,
a redistricting algorithm must therefore aim to simultaneously be fair to each
party with respect to its own reported data. Our main theoretical result is
that, surprisingly, the geometric target is always feasible with respect to
arbitrarily diverging data sets on how voters are distributed.
  Any standard for fairness is only useful if it can be readily satisfied in
practice. Our empirical results, which use real election data and maps of six
US states, demonstrate that the geometric target is always feasible, and that
imposing it as a fairness constraint comes at almost no cost to three
well-studied optimization objectives.
