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

We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre Bárány. Given a positive integer $k$ and $d$ nice measures $μ_1, μ_2,..., μ_d$ 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 $μ_i (C_j) = 1$ for all $i,j$. If $k=2$ this gives the ham sandwich Theorem.

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