A Medial-Axis-Based Measure of District Compactness

Abstract

An essential question for democracy is how to rigorously determine the likelihood that a congressional map has been gerrymandered. A number of state constitutions require districting plans to be "compact", yet no set definition of compactness exists, other than an oft-cited sentiment that you "know it when you see it". We introduce a novel compactness measure based on a geometric structure known as the medial axis, which has strong ties to the science of how humans perceive and process shapes. As such, we argue that our metric performs well as a mathematical quantification of ``"knowing it when you see it”. We compare our medial-axis-based measure to a recent machine-learning-based compactness metric introduced by Kaufman, King, and Komisarchik. Specifically, we examine the performance of our measure and theirs in several case studies, including two states whose districting plans have been extensively covered in the media as well as the entire 2016 congressional district map.