Geodesic
Separation of $(n+1)$-families of sets in general position in $\bold R^n$
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 4, pp. 743-748 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${\bold R}^n$, is generalized. Finally we prove the following theorem: If $\{A_1,A_2,\dots,A_{n+1}\}$ is a family of compact convexly connected sets in general position in ${\bold R}^n$, then for each proper subset $I$ of $\{1,2,\dots,n+1\}$ the set of hyperplanes separating $\cup\{A_i: i\in I\}$ and $\cup\{A_j: j\in \overline{I}\}$ is homeomorphic to $S_n^+$.
In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${\bold R}^n$, is generalized. Finally we prove the following theorem: If $\{A_1,A_2,\dots,A_{n+1}\}$ is a family of compact convexly connected sets in general position in ${\bold R}^n$, then for each proper subset $I$ of $\{1,2,\dots,n+1\}$ the set of hyperplanes separating $\cup\{A_i: i\in I\}$ and $\cup\{A_j: j\in \overline{I}\}$ is homeomorphic to $S_n^+$.
Classification : 47H10, 52A37
Keywords: family of sets in general position; convexly connected sets; Fan-Glicksberg-Kakutani fixed point theorem 
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     author = {Balaj, Mircea},
     title = {Separation of $(n+1)$-families of sets in general position in $\bold R^n$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {743--748},
     year = {1997},
     volume = {38},
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     zbl = {0946.52002},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1997_38_4_a9/}
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Balaj, Mircea. Separation of $(n+1)$-families of sets in general position in $\bold R^n$. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 4, pp. 743-748. http://geodesic.mathdoc.fr/item/CMUC_1997_38_4_a9/