Discrete & Computational Geometry

, Volume 43, Issue 4, pp 876–892 | Cite as

Uneven Splitting of Ham Sandwiches

Article

Abstract

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 \(\alpha_{i}=\frac{1}{2}\) for all i, then such a hyperplane always exists.

In this paper we give sufficient criteria for the existence of H for general α i ∈[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 μ i (H i + )=α 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 μ i (H +)=α i for all i. This gives rise to several corollaries; for instance, if the supports of μ 1,…,μ n are bounded and can be separated by hyperplanes, then H exists for any choice of α 1,…,α n ∈[0,1]. We also obtain results that can be applied if the supports of μ 1,…,μ n overlap.

Keywords

Partitions of masses Ham sandwich theorem Poincaré–Miranda theorem Hyperplanes Separability 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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