Slicing convex sets and measures by a hyperplane

Abstract

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in ${\mathbb{R}}_d$ and numbers ${\alpha }_1,...,{\alpha }_d\in [0,1]$, we give a sufficient condition for existence and uniqueness of an (oriented)halfspace H with Vol($H\cap K_i$)= $α_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.