Abstract
We introduce the convexity ratio, CR(P), which indicates how close a
simple closed polygon P is to being convex. This is accomplished by considering the
ratio of the area of a largest convex set (the endogon) contained in P to the area of the
convex hull of P. Algorithms for determining the convex hull and for determining the
endogon are exhibited. After defining when such polygons are nearly convex we then
use this result to decide when legislative districts are nicely shaped. We show that this
method for measuring the shape of legislative districts is better than, or as good as,
other techniques in the literature. This is accomplished by pointing out flaws in the
other methods and by examining examples of district shapes found in the literature.
