Uneven Splitting of Ham Sandwiches

Abstract

Let m_1,...,m_n be continuous probability measures on R^n and a_1,...,a_n in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H^+ has m_i(H^+)=a_i for all i in [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if a_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 a_i in [0,1]. Let f_1,...,f_n:S^{n-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 m_i(H_i^+)=a_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 m_i(H^+)=a_i for all i. This gives rise to several corollaries, for instance if the supports of m_1,...,m_n are bounded and can be separated by hyperplanes, then H exists for any choice of a_1,...,a_n in [0,1]. We also obtain results that can be applied if the supports of m_1,...,m_n overlap.