Abstract
In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. We evaluate these results empirically on United States congressional districts and demonstrate that this order-reversal does occur in practice with respect to commonly used projections. Furthermore, the Reock score ordering in particular appears to be quite sensitive to the choice of map projection.