Distance between vertices of lattice polytopes

Abstract

A lattice (d,k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers ranging between 0 and k. We consider the largest possible distance $$\delta $$(d,k) between two vertices in the edge-graph of a lattice (d,k)-polytope. We show that $$\delta $$(5,3) and $$\delta $$(3,6) are equal to 10. This substantiates the conjecture whereby $$\delta $$(d,k) is achieved by a Minkowski sum of lattice vectors.