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My question is motivated by the definition of electoral districts.

One way of making gerrymandering more difficult would be to require that districts be convex. A broader approach to the problem was asked at Mathematics solution for Gerrymandering problem?

However, this is probably too restrictive and difficult to achieve in face of other restrictions. So, a better solution would be to have districts "as convex as possible". For that to make sense, one would like to have a measure of how close to convex a non-convex set is, and then try to optimize some total non-convexity of a district subdivision. Has any such measure ever been studied?

The application I am interested in is $2$-dimensional, so specialized definitions would be welcome.

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