  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.
