Discrete & Computational Geometry

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

Uneven Splitting of Ham Sandwiches



Let μ1,…,μn be continuous probability measures on ℝn and α1,…,αn∈[0,1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has μi(H+)=αi 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 αi∈[0,1]. Let f1,…,fn:Sn−1→ℝn denote auxiliary functions with the property that for all i, the unique hyperplane Hi with normal v that contains the point fi(v) has μi(Hi+)=αi. Our main result is that if Im f1,…,Im fn are bounded and can be separated by hyperplanes, then there exists a hyperplane H with μi(H+)=αi for all i. This gives rise to several corollaries; for instance, if the supports of μ1,…,μn are bounded and can be separated by hyperplanes, then H exists for any choice of α1,…,αn∈[0,1]. We also obtain results that can be applied if the supports of μ1,…,μn overlap.


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


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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