Uneven Splitting of Ham Sandwiches

TL;DR

This paper extends the ham sandwich theorem by providing criteria for the existence of hyperplanes that evenly split multiple measures with arbitrary proportions, even when supports overlap or are unbounded.

Contribution

It introduces sufficient conditions based on auxiliary functions and separation of their images, broadening the applicability of measure-splitting hyperplanes beyond equal division.

Findings

Existence of hyperplanes for arbitrary measure proportions under bounded and separable support conditions.

Conditions involving auxiliary functions that determine hyperplane existence.

Applicability to overlapping measure supports.

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.