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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.Beneš (Praha)
##### MSC:
 52A37 Other problems of combinatorial convexity
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