Uneven Splitting of Ham Sandwiches
نویسنده
چکیده
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 αi = 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(H i ) = 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.
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ورودعنوان ژورنال:
- Discrete & Computational Geometry
دوره 43 شماره
صفحات -
تاریخ انتشار 2010