Slicing convex sets and measures by a hyperplane

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,\mathrm{.}\mathrm{.}\mathrm{.},K_d$ in ${\mathbb{R}}_{\mathbb{d}}$ and numbers ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_d\in [0,1]$, 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.