Motivated exposition of the proof of the Tverberg Theorem

Abstract: We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers $d,r$ any $(d+1)(r-1)+1$ points in $\mathbb R^d$ can be decomposed into $r$ groups such that all the $r$ convex hulls of the groups have a common point. The proof is by well-known reduction to the B\'ar\'any Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $\mathbb R^d\subset\mathbb R^{d+1}$, of vectors $\varphi_{j,i}$ and statement of the Bara\'ny Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.