ham sandwich theorem HamSandwichTheorem 2003-10-13 00:31:53 2003-11-24 16:00:09 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} Let $A_1,\ldots,A_m$ be measurable bounded subsets of $\mathbb{R}^m$. Then there exists an $(m-1)$-dimensional hyperplane which \PMlinkescapetext{divides} each $A_i$ into two subsets of equal measure. This theorem has such a colorful \PMlinkescapetext{name} 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 \PMlinkescapetext{cut} to simultaneously \PMlinkescapetext{cut} all three objects exactly in half.