Uneven Splitting of Ham Sandwiches

Let mu(1), ... , mu(n) be continuous probability measures on R-n and alpha(1), ... , alpha(n) is an element of [0, 1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has mu(i)(H+) = alpha(i) for all i is an element of [n]? It is well known that such a hyper-plane does not exist in general. The famous Ham Sandwich Theorem states that if alpha(i) = 1/2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general alpha(i) is an element of [0, 1]. Let f(1), ... , f(n) : Sn-1 -> R-n denote auxiliary functions with the property that for all i, the unique hyperplane H-i with normal v that contains the point f(i)(v) has mu(i)(H-i(+)) = alpha(i). Our main result is that if Im f(1), ... , Im f(n) are bounded and can be separated by hyperplanes, then there exists a hyperplane H with mu(i)(H+) = alpha(i) for all i. This gives rise to several corollaries; for instance, if the supports of mu(1), ... , mu(n) are bounded and can be separated by hyperplanes, then H exists for any choice of alpha(1), ... , alpha(n) is an element of [0, 1]. We also obtain results that can be applied if the supports of mu(1), ... , mu(n) overlap.