Balanced convex partitions of measures in \(\mathbb R^{d}\). (English) Zbl 1267.28005

Summary: We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer \(k\) and \(d\) nice measures \(\mu _{1},\mu _{2},\dotsc ,\mu _{d}\) in \(\mathbb R^{d}\) such that \(\mu _{i}(\mathbb R^{d})=k\) for all \(i\), there is a partition of \(\mathbb R^{d}\) into \(k\) interior-disjoint convex parts \(C_{1},C_{2},\dotsc ,C_{k}\) such that \(\mu _{i} (C_{j})=1\) for all \(i\), \(j\). If \(k=2\), this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.


28A75 Length, area, volume, other geometric measure theory
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