Related papers: Reflexive polytopes of higher index and the number…
It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity,…
Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on…
We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry…
It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every $(0,1)$-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large…
The Ehrhart function $L_P(t)$ of a polytope $P$ is usually defined only for integer dilation arguments $t$. By allowing arbitrary real numbers as arguments we may also detect integer points entering (or leaving) the polytope in fractional…
Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite…
We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $area(\Delta) + area(\Delta^*) = 6$ for reflexive $\Delta$. The identity is equivalent to the stringy Libgober-Wood…
A lattice polytope $\mathcal{P}$ is called reflexive if its dual $\mathcal{P}^\vee$ is a lattice polytope. The property that $\mathcal{P}$ is unimodularly equivalent to $\mathcal{P}^\vee$ does not hold in general, and in fact there are few…
A classification of discrete polymatroids whose independence polytopes are reflexive will be presented.
The reflexive dimension refldim(P) of a lattice polytope P is the minimal d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of…
In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight…
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties.…
Given arbitrary integers $d$ and $r$ with $d \geq 4$ and $1 \leq r \leq d + 1$, a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ with ${\rm depth} K[\mathcal{P}] = r$ for which its dual polytope $\mathcal{P}^\vee$ is…
Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes…
Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of…
We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed…
Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in "duality theories". In this paper we study and determine reflexive functors and we give many…
We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer-Scott axiom of combinatory models, which indeed allows us to characterise an equationally definable…