Redistricting without gerrymandering, utilizing the convexity ratio, and other applications to business and industry

Abstract

We exhibit the convexity ratio of voting districts in many states of the USA, which have had their plans challenged. The convexity ratio confirms that these states have likely been gerrymandered. We then redistrict the largest of these states, Texas, avoiding gerrymandering by starting with counties and their populations and only adding or deleting municipalities as necessary. The voting districts themselves are made as nicely shaped as possible, using the convexity ratio as a guide. Those districts with immovable boundaries that cause a poorly shaped designation, a topic of much current research, are dealt with appropriately in the context of the convexity ratio. Algorithms for finding the convexity ratio and designing non-gerrymandered districts are shown. Other more probabilistic convexity measures are discussed. Finally, we comment on how the convexity ratio can be used in other countries, both politically and in other contexts, for example, in territory design for geomarketing and on the electrical or police districting problem.