The depth of a reflexive polytope

TL;DR

This paper constructs reflexive polytopes of any given dimension and depth, with normal duals, advancing understanding of the relationship between polytope depth and dual normality.

Contribution

It provides a method to construct reflexive polytopes with prescribed depth and normal duals, filling a gap in the classification of such polytopes.

Findings

Constructed reflexive polytopes for all specified depths and dimensions.

Showed the dual polytopes are normal in these constructions.

Extended the understanding of the relationship between polytope depth and dual normality.

Abstract

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 normal will be constructed, where $K[\mathcal{P}]$ is the toric ring of $\mathcal{P}$.