I'm looking within my PhD at atm at decomposing a random non-convex subset of the Euclidean Plane into a union of n convex sets, particularly hoping that the these sets (that from the overall non-convex union) don't overlap.
I've found that an m-convex set can be decomposed into phi(m) subsets (which is a finite integer), see Breen 1977 Pacific Journal of Maths "Decomposition for Closed Non-Planar m-Convex Sets". [It is not clear whether these sets overlap.]
However, if a I take a random non-convex set (basically a stochastic geometric object which is non-convex), how do I determine the 'm' of its 'm-convexity'? (an m-convex set is so defined as a set where for any m points in the set, at least one of the line segments between any two of the said points is in the set).
Any help? Regards.