2008/07/17 by Felix Breuer · 5 citations
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Bounded function #Existential quantification #Hyperplane #Limits and Structures in Graph Theory #Point (geometry) #Property (philosophy) #math.CO #msc:52A37
paper · pdf · doi:10.1007/s00454-009-9161-7
published in Discrete & Computational Geometry 43(4), 876-892 (Springer Science+Business Media) · 18 pages, 2 figures
arxiv created 2008/07/17 · openalex publication_date 2009/04/14 · arxiv updated 2010/04/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let m1,...,mn be continuous probability measures on Rn and a1,...,an in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has mi(H+)=ai 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 ai=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 ai in [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 mi(Hi+)=ai. 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 mi(H+)=ai for all i. This gives rise to several corollaries, for instance if the supports of m1,...,mn are bounded and can be separated by hyperplanes, then H exists for any choice of a1,...,an in [0,1]. We also obtain results that can be applied if the supports of m1,...,mn overlap.