The diameter of lattice zonotopes
Copyright policy )The diameter of lattice zonotopes
We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers $k$, the largest possible diameter of a lattice zonotope contained in the hypercube $[0,k]^d$ is uniquely achieved by a primitive zonotope. As a consequence, we obtain that this largest diameter grows like $k^{d/(d+1)}$ up to an explicit multiplicative constant, when $d$ is fixed and $k$ goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in $[0,k]^d$.
12 pages
Distance in graphs, Metric Geometry (math.MG), primitive zonotope, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], \(n\)-dimensional polytopes, Mathematics - Metric Geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Lattices and convex bodies (number-theoretic aspects), FOS: Mathematics, lattice zonotope, Mathematics - Combinatorics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Combinatorics (math.CO), diameter
Distance in graphs, Metric Geometry (math.MG), primitive zonotope, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], \(n\)-dimensional polytopes, Mathematics - Metric Geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Lattices and convex bodies (number-theoretic aspects), FOS: Mathematics, lattice zonotope, Mathematics - Combinatorics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Combinatorics (math.CO), diameter
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