Balanced Convex Partitions of Measures in ${\mathbb{R}}^d$

Abstract

We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre Bárány. Given a positive integer $k$ and $d$ nice measures ${\mu }_1,{\mu }_2,...,{\mu }_d$ in ${\mathbb{R}}^d$ such that $μ_i (\mathds{R}^d) = k$ for all $i$, there is a partition of ${\mathbb{R}}^d$ in $k$ interior-disjoint convex parts $C_1,C_2,...,C_k$ such that ${\mu }_i(C_j)=1$ for all $i,j$. If $k=2$ this gives the ham sandwich Theorem.

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