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($H \cap K_i$)= $\alpha_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.