Diameter, Decomposability, and Minkowski Sums of Polytopes

Antoine Deza (1) , Lionel Pournin (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 LIPN - Laboratoire d'Informatique de Paris-Nord

Abstract We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$ , we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$ .

Domaines: Géométrie algorithmique [cs.CG]; Combinatoire [math.CO]; Optimisation et contrôle [math.OC]; Mathématique discrète [cs.DM]

Antoine Deza, Lionel Pournin. Diameter, Decomposability, and Minkowski Sums of Polytopes. Canadian Mathematical Bulletin, 2019, 62 (4), pp.741-755. ⟨10.4153/S0008439518000668⟩. ⟨hal-03410029⟩