A medial axis-based compactness measure vs. the eye test for detecting gerrymandering*

Abstract: We, like many other researchers, are interested in providing a mathematical basis for determining the likelihood of whether a congressional or legislative district has been gerrymandered. A judicial precedent exists that one cannot define an exact notion of a district's degree of “compactness” but you “know it when you see it.” Our paper explores the effectiveness of a medial axis-based compactness measure involving the extended distance function that can be calculated directly on the medial axis. We make the case that this measure both objectively performs well in identifying gerrymandered districts and is rooted in an accurate representation of how humans perceive and process shape. We compare our medial axis-based compactness score to other popular methods of assessing compactness, and we analyze the performance of our model in several case studies involving states whose districting plans have been extensively covered in the media.

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