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Separation of \((n+1)\)-families of sets in general position in \({\mathbf R}^n\). (English) Zbl 0946.52002
In the paper separations of members of an \((n+1)\)-family of sets in general position in \(R^n\) are studied. Previous achievements of the author are substantially improved. Using the tool Fan-Glicksberg-Kakutani fixed point theorem the following result is finally proved: If \(\{A_1,A_2,\ldots ,A_n\}\) is a family of compact convexly connected sets in general position in \(R^n\), then for each proper subset \(I\) of \(\{1,2,\ldots ,n+1\}\) and its complement \(\bar I\) the set of hyperplanes separating \(\cup \{A_i,i\in I\}\) and \(\cup \{A_j, j\in \bar I\}\) is homeomorphic to \(S_+=\{x\in R^{n+1}; \|x\|=1\), \(x_i \geq 0\), \(1\leq i \leq n+1\}\).
Reviewer: V.BeneŇ° (Praha)
MSC:
52A37 Other problems of combinatorial convexity
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