  We model the societal task of redistricting political districts as a
partitioning problem: Given a set of $n$ points in the plane, each belonging to
one of two parties, and a parameter $k$, our goal is to compute a partition
$\Pi$ of the plane into regions so that each region contains roughly $\sigma =
n/k$ points. $\Pi$ should satisfy a notion of ''local'' fairness, which is
related to the notion of core, a well-studied concept in cooperative game
theory. A region is associated with the majority party in that region, and a
point is unhappy in $\Pi$ if it belongs to the minority party. A group $D$ of
roughly $\sigma$ contiguous points is called a deviating group with respect to
$\Pi$ if majority of points in $D$ are unhappy in $\Pi$. The partition $\Pi$ is
locally fair if there is no deviating group with respect to $\Pi$.
  This paper focuses on a restricted case when points lie in $1$D. The problem
is non-trivial even in this case. We consider both adversarial and ''beyond
worst-case" settings for this problem. For the former, we characterize the
input parameters for which a locally fair partition always exists; we also show
that a locally fair partition may not exist for certain parameters. We then
consider input models where there are ''runs'' of red and blue points. For such
clustered inputs, we show that a locally fair partition may not exist for
certain values of $\sigma$, but an approximate locally fair partition exists if
we allow some regions to have smaller sizes. We finally present a
polynomial-time algorithm for computing a locally fair partition if one exists.
