Boundary optimization for rough sets

Let n and m be integers, n>m≥2, and let A={A1,…,Am} be a partition of [n], where [n]={1,…,n}. For a subset X of [n], its A-boundary region A(X) is defined to be the union of those blocks Ai of A for which Ai∩X≠∅ and Ai∩([n]∖X)≠∅. Partitions A of [n] into m parts are determined such that, in a first case, the expected absolute cardinality and, in a second case, the expected relative cardinality of the A-boundary region of a randomly chosen subset of [n] is minimal and maximal, respectively. The problem can be reduced to an optimization problem for integer partitions of n. In the most difficult case, the concave-convex shape of the corresponding weight function as well as several other inequalities are proved using an integral representation of the weight function. In a modified third setting, there is an analogon to the AZ-identity. The study is motivated by rough set theory.