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
Canonical name	HamSandwichTheorem
Date of creation	2013-03-22 13:59:43
Last modified on	2013-03-22 13:59:43
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 54C99
Related topic	BorsukUlamTheorem