ham sandwich theorem

Let $A_1,\mathrm{\dots },A_m$ be measurable bounded subsets of ${ℝ}^m$. Then there exists an $(m-1)$-dimensional hyperplane which each $A_i$ into two subsets of equal measure.

This theorem has such a colorful because in the case $m=3$ it can be viewed as cutting a ham sandwich in half. For example, $A_1$ and $A_3$ could be two pieces of bread and $A_2$ a piece of ham. According to this theorem it is possible to make one to simultaneously all three objects exactly in half.