Uneven Splitting of Ham Sandwiches

摘要

Let μ ₁,…,μ _n be continuous probability measures on ℝ n and α ₁,…,α _n ∈[0,1]. When does there exist an oriented hyperplane H such that the positive half-space H + has μ _i (H +)=α _i for all i∈[n]? It is well known that such a hyperplane does not exist in general. The famous Ham Sandwich Theorem states that if for all i, then such a hyperplane always exists.