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Discrete and Computational Geometry
October 23, 2013 13:00, Moscow, IITP RAS, Bolshoy Karetny per. 19, room 307
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Quantitative theorems on covering dimension and toric geometry
R. N. Karasev |
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This page: | 164 |
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Abstract:
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.
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