Mixed polytopes

Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.