# Unconditional reflexive polytopes

## TL;DR

This paper explores unconditional lattice polytopes, especially reflexive ones, characterizing them via perfect graphs, and provides explicit algebraic and combinatorial descriptions including Gr"obner bases for certain classes.

## Contribution

It introduces a characterization of unconditional reflexive polytopes through perfect graphs and develops explicit algebraic tools for their analysis.

## Key findings

- Characterization of unconditional reflexive polytopes via perfect graphs
- Explicit description of Gr"obner bases for polytopes from posets
- Construction methods for Gale-dual pairs of unconditional polytopes

## Abstract

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. 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 Gr\"obner bases for unconditional reflexive polytopes coming from partially ordered sets