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

TL;DR

This paper generalizes the ham sandwich theorem by proving that for any positive integer k and d measures of equal total measure in , there exists a convex partition into k parts each with equal measure for all measures.

Contribution

It proves a conjecture by Be1re1ny extending the ham sandwich theorem to partitions into k convex parts with equal measures.

Findings

Established existence of convex partitions for measures in 

Generalized the ham sandwich theorem to arbitrary k

Confirmed Be1re1ny's conjecture

Abstract

We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre B\'ar\'any. Given a positive integer $k$ and $d$ nice measures $\mu_1, \mu_2,..., \mu_d$ in $\mathbb{R}^d$ such that $\mu_i (\mathds{R}^d) = k$ 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.