# Has a degree of convexity ever been defined?

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.

• Distant relative: mathoverflow.net/q/187819/91764 Jul 17, 2020 at 13:00
• Related: math.stackexchange.com/q/38772 Jul 17, 2020 at 17:30
• A less distant relative: Willmore energy Jul 19, 2020 at 0:49
• I was thinking that curvature would have a role to play - requiring a smooth boundary is not a problem, and minimizing Willmore energy (under additional constraints) seems to mold the shape into a desirable form. Thanks! Jul 19, 2020 at 10:37