Geometric spaces from arbitrary convex polytopes

TL;DR

This paper introduces a method to construct geometric spaces from any convex polytope, extending the concept of toric varieties to a broader class of polytopes with similar stratification properties.

Contribution

It generalizes the construction of toric varieties to arbitrary convex polytopes, providing new geometric spaces with stratification akin to classical toric varieties.

Findings

Constructed geometric spaces from arbitrary convex polytopes.

Spaces exhibit stratification similar to toric varieties.

Extension of toric geometry concepts to non-rational polytopes.

Abstract

We associate a geometric space to an arbitrary convex polytope. Our construction parallels the construction by D. Cox of a toric variety as a GIT quotient. The spaces that we obtain are endowed with a natural stratification and perfectly mimic the features of toric varieties associated to rational convex polytopes.