Diameter, decomposability, and Minkowski sums of polytopes

TL;DR

This paper explores how Minkowski sums influence the diameter and decomposability of polytopes, establishing bounds and conditions for polytope decomposition with sharpness proofs.

Contribution

It provides new bounds on the diameter of Minkowski sums of polytopes and characterizes polytope decomposability in terms of vertex counts.

Findings

Diameter bounds are sharp and relate to summand properties.

Polytope P can be decomposed with Q iff vertex counts match.

Results connect polytope structure with Minkowski sum properties.

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.