Sperner type lemma for quadrangulations (Q268063)

Let \(S\) be a \(d\)-dimensional simplex with vertices \(v_1, \dots, v_{d+1}\). Let \(T\) be a triangulation of \(S\). Suppose that to each vertex of \(T\) a unique label from the set \(\{1, 2, \dots, d+1\}\) is assigned. A labelling \(L\) is called Sperner if the vertices are labelled in such a way that a vertex of \(T\) belonging to the interior of a face \(F\) of \(S\) can only be labelled by \(k\) if \(v_k\) is on \(S\). Sperner's Lemma is a discrete analog of the Brouwer fixed point theorem and states that every Sperner labelling of a triangulation of a \(d\)-dimensional simplex contains a cell labelled with a complete set of labels: \(\{1, 2, \dots, d+1\}\).NEWLINENEWLINEIn the paper under review, the author presents a generalization of this lemma by considering the quadrangulations instead of triangulations.