Quantitative theorems on covering dimension and toric geometry

It is a classical fact that the covering dimension of $\mathbb R^n$ equals $n$ and there are classical “quantitative” versions if this fact: the Lebesgue theorem and the Knaster–Kuratowski–Mazurkiewicz theorem. Informally, they assert that if the covering multiplicity is at most $n$ then some of the covering sets must be definitely large.
Discrete analogues of these results are known as the Sperner lemma and the HEX lemma.
It turns out that these results have a simple explanation in terms of the toric varieties, corresponding to the cube and the simplex. Moreover, the toric approach allows to prove some generalizations of these theorems. In particular, a topological version of the center point theorem also follows.