Generalized flatness constants, spanning lattice polytopes, and the Gromov width

Abstract

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

Moduli spaces of Delzant polytopes and symplectic toric manifolds

“…Faces of dimension dim(P ) − 1 are called facets. 2 A (convex) cone in V is any subset that is closed under addition and multiplication by nonnegative scalars. A polyhedral cone is a cone that is also a polyhedron.…”

“…(1) The original rays of Σ have valency at least six in Σ ′′ and hence in Σ ′ , so they cannot be blown down. (2) The rays introduced in the blow-ups from Σ to Σ ′′ have valency three in Σ ′′ , hence in Σ ′ . Let α be one of them, and let β, γ, δ be its neighbors in Σ ′′ , so that α = β + γ + δ.…”

“…In part (2), the fact that D(n) is neither complete nor locally compact is proved in [45,Theorem 2] for n = 2 and the proof carries over for higher n.…”

“…If µ happens to be zero, we are again violating one of the inequalities in part (1). If µ > 0 then we can assume without loss of generality that µ = 1 and we are violating the inequality i λ i b i b j in part (2).…”

“…We note that other moduli spaces of polygons have been studied by a number of authors, for example see Hausmann-Knutson [25,26] and . Also, the idea of using the Delzant correspondence to solve problems in symplectic geometry by solving problems in discrete geometry is not new, see for instance [2,42,43] and the references therein. Finally, we note that having these moduli space structures is useful to study the variation of properties (such as continuity) of functions on these moduli spaces, see for example Figalli-Pelayo [17] and Figalli-Palmer-Pelayo [18].…”