Toric Varieties vs. Horofunction Compactifications of Polyhedral Norms

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.