Abstract
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. We investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets.
Highlights
A d-dimensional convex lattice polytope P ⊂ Rd is called reflexive if its polar dual P∗ is again a lattice polytope
Reflexive polytopes were introduced by Batyrev [6] in the context of mirror symmetry as a reflexive polytope and its dual give rise to a mirrordual pair of Calabi–Yau manifolds; cf. [22]
The purpose of this paper is to study a class of reflexive polytopes motivated by convex geometry and relate it to combinatorics
Summary
A d-dimensional convex lattice polytope P ⊂ Rd is called reflexive if its polar dual P∗ is again a lattice polytope. The purpose of this paper is to study a class of reflexive polytopes motivated by convex geometry and relate it to combinatorics. Unconditional convex bodies, for example, arise as unit balls in the theory of Banach spaces with a 1-unconditional basis. We mention that the Mahler conjecture is known to hold for unconditional convex bodies; see Sect. We show in Theorems 4.6 and 4.9 that an unconditional polytope P is reflexive if and only if P = UPG for some unique perfect graph G. This implies that unconditional reflexive polytopes have regular, unimodular triangulations.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
