Minkowski addition of convex polytopes

This note summarizes recent results from computational geometry which determine complexity of computing Minkowski sum of k convex polytopes in R, which are represented either in terms of facets or in terms of vertices. In particular, it is pointed out for which cases there exists an algorithm which runs in polynomial time. The note is based on papers of Gritzmann and Sturmfels [6] and Komei Fukuda [3]. An algorithm which aims at reducing the complexity of obtaining minimal representation of polytopes given by a set of inequalities is presented as well. 1 Polytopes Definition 1 (polyhedron) A convex set Q ⊆ R given as an intersection of a finite number of closed half-spaces Q = {x ∈ R | Qx ≤ Q}, (1) is called polyhedron. Definition 2 (polytope) A bounded polyhedron P ⊂ R P = {x ∈ R | P x ≤ P }, (2) is called polytope. It is obvious from the above definitions that every polytope represents a convex, compact (i.e., bounded and closed) set. We say that a polytope P ⊂ R, P = {x ∈ R | P x ≤ P } is full dimensional if ∃x ∈ R : P x < P . Furthermore, if ‖(P )i‖ = 1, where (P )i denotes i-th row of a matrix P , we say that the polytope P is normalized. One of the fundamental properties of a polytope is that it can also be described by its vertices