Distance between vertices of lattice polytopes

Anna Deza , Antoine Deza , Zhongyan Guan , Lionel Pournin (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

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 δ (d, k) between two vertices in the edge-graph of a lattice (d, k)-polytope. We show that δ (5, 3) and δ (3, 6) are equal to 10. This substantiates the conjecture whereby δ (d, k) is achieved by a Minkowski sum of lattice vectors.

Mots clés: Lattice Polytope; Diameter; Minkowski Sum

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

Anna Deza, Antoine Deza, Zhongyan Guan, Lionel Pournin. Distance between vertices of lattice polytopes. Optimization Letters, 2020, 14 (2), pp.309-326. ⟨10.1007/s11590-018-1338-7⟩. ⟨hal-03412184⟩