Convexity Coefficient

The convexity coefficient  of a region  is the probability that the line segment connecting two random points in  is contained entirely within . For a convex region, .

For  a subset of , let the area of the visible region of a point  be denoted , and let the area of  be denoted . Then

(Hodge et al. 2010).

The convexity coefficient is in general hard to compute exactly for concave regions even of simple shape. One closed form is that of an annulus with inner radius  and outer radius , which has

(Hodge et al. 2010).

REFERENCES:

Hodge, J.; Marshall, E.; and Patterson, G. "Gerrymandering and Convexity." Coll. Math. J. 41, 312-324, 2010.