Tverberg定理证明的启发性论述
Motivated exposition of the proof of the Tverberg Theorem
摘要：
我们提出了一个有动机的阐述以下Tverberg定理的证明：对于每一个整数$d,r$任何$(d+1)(r-1)+1$点在$\mathbb R^d$可以分解成$r$组，使所有的$r$凸包的组有一个公共点。这个证明是由著名的归约到Bárány定理。然而，我们的阐述更容易理解，因为额外的构造（嵌入$\mathbb R^d\subset\mathbb R^{d+1}$的构造、向量$φ_{j,i}$的构造和Baraubrey定理的陈述）不是以非动机的方式预先引入的，而是自然地出现在试图构造所需分解的过程中。这种尝试是基于重写向量之间的几个等式作为一个更高维度的向量之间的等式。
英文摘要：
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 Bárány Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $\mathbb R^d\subset\mathbb R^{d+1}$, of vectors $φ_{j,i}$ and statement of the Barańy 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.
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