Question

Mostly need help with part b:

(a) * Consider a tetrahedron T = A1 A2A3A4 in the 3-dimensional space and some subdivision of T into small tetrahedra, such that each face of each small tetrahedron either lies on a face of the big tetrahedron or is also a face of another small tetrahedron. Let us label the vertices of the small tetrahedra by labels 1. 2, 3, 4, in such a way that the vertex Ai gets i, the edge A,Aj only contains vertices labeled i and j, and the face A, AjAk has only labels i,j, and k. Prove that there exists a small tetrahedron labeled 1, 2, 3, 4 (b) Formulate and prove a 3-dimensional version of Brouwer's fixed point theorem (about continuous mappings of a tetrahedron into itself)

Answer 1

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