Toric Varieties vs. Horofunction Compactifications of Polyhedral Norms

TL;DR

This paper reveals a natural correspondence between toric varieties, polyhedral norms, and horofunction compactifications, bridging algebraic geometry, convex analysis, and metric geometry.

Contribution

It establishes a geometric 1-1 correspondence between projective toric varieties and horofunction compactifications of Euclidean space with rational polyhedral norms.

Findings

Toric varieties are topologically modeled in a way that relates to horofunction compactifications.

A direct and explicit connection is made between algebraic geometry, convex analysis, and metric geometry.

The correspondence provides new insights into the structure of polyhedral norms and their geometric compactifications.

Abstract

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geometry, normed spaces in convex analysis, and horofunction compactifications in metric geometry are directly and explicitly related.