# ham sandwich theorem

Let $A_{1},\ldots,A_{m}$ be measurable bounded subsets of $\mathbb{R}^{m}$. 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.

Title ham sandwich theorem HamSandwichTheorem 2013-03-22 13:59:43 2013-03-22 13:59:43 mathcam (2727) mathcam (2727) 6 mathcam (2727) Theorem msc 54C99 BorsukUlamTheorem