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2015-03-17 Nonlinear Seminar

Piotr Maćkowiak
A combinatorial lemma ant it's consequences

During this seminar we shall introduce a combinatorial lemma stating the existence for particular triangulations of the $$ (n-1) $$-dimensional simplex with a particular labeling of vertices of such triangulations with numbers $$ \{0, 1,\ldots , n\} $$ of a chain of simplices connecting a "vartex" with a "base" such that for any simplex from that chain each element of the set $$  \{1,\ldots , n-1\}  $$ is a label of some vertex of that simplex. We shall show, that (simple) consequences of this lemma are:

  1. the existence of zeros of the multidimensional function of excess demand
  2. the existence of continuum zeros of the parametrised multidimensional function of excess demand
  3. the well-known Sperner lemma