  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.
