A Single Compactness Measure for Legislative Redistricting.

Legislative districts in the 50 states will be redrawn following the completion of the 2010 United States census. Thirty-five require that districts be compact, which is believed to make gerrymandering - designing legislative districts so as to advantage one party - more difficult (Chambers and Miller, 2009). Geographic compactness also affects how convenient it is for legislators to travel within their districts to meet with their constituents, how easy it is for election officials to administer elections, and for constituents to participate in the legislative process. When districts are compact neighbors having similar interests are more likely to live in the same district. There are now more than a dozen competing proposals for evaluating the shapes of legislative districts. This article clears away some confusion over area compactness measures by demonstrating that nine proposed measures of district compactness fail to reliably measure compactness, while at least four other proposed measures of district compactness are equivalent to one simpler measure of area compactness. By providing counterexamples, we show that nine proposed measures of area compactness assign the same value to shapes having different compactness levels, and thus are unreliable measures. Then, we provide a mathematical induction proof showing that all area-to-square-of-perimeter measures (or its reciprocal or square root) rank the compactness of any two sets of redistricting plans in the exact same order. This kind of index of compactness is conceptually and computationally simpler than others and should be used. [ABSTRACT FROM AUTHOR]