# Separation of $(n+1)$-families of sets in general position in ${\mathbf{R}}^{n}$

Commentationes Mathematicae Universitatis Carolinae (1997)

- Volume: 38, Issue: 4, page 743-748
- ISSN: 0010-2628

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topBalaj, Mircea. "Separation of $(n+1)$-families of sets in general position in $\mathbf {R}^n$." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 743-748. <http://eudml.org/doc/248068>.

@article{Balaj1997,

abstract = {In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in $\{\mathbf \{R\}\}^n$, is generalized. Finally we prove the following theorem: If $\lbrace A_1,A_2,\dots ,A_\{n+1\}\rbrace $ is a family of compact convexly connected sets in general position in $\{\mathbf \{R\}\}^n$, then for each proper subset $I$ of $\lbrace 1,2,\dots ,n+1\rbrace $ the set of hyperplanes separating $\cup \lbrace A_i: i\in I\rbrace $ and $\cup \lbrace A_j: j\in \overline\{I\}\rbrace $ is homeomorphic to $S_n^+$.},

author = {Balaj, Mircea},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {family of sets in general position; convexly connected sets; Fan-Glicksberg-Kakutani fixed point theorem; combinatorial geometry; convexly connected sets; family of sets in general position},

language = {eng},

number = {4},

pages = {743-748},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Separation of $(n+1)$-families of sets in general position in $\mathbf \{R\}^n$},

url = {http://eudml.org/doc/248068},

volume = {38},

year = {1997},

}

TY - JOUR

AU - Balaj, Mircea

TI - Separation of $(n+1)$-families of sets in general position in $\mathbf {R}^n$

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1997

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 38

IS - 4

SP - 743

EP - 748

AB - In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${\mathbf {R}}^n$, is generalized. Finally we prove the following theorem: If $\lbrace A_1,A_2,\dots ,A_{n+1}\rbrace $ is a family of compact convexly connected sets in general position in ${\mathbf {R}}^n$, then for each proper subset $I$ of $\lbrace 1,2,\dots ,n+1\rbrace $ the set of hyperplanes separating $\cup \lbrace A_i: i\in I\rbrace $ and $\cup \lbrace A_j: j\in \overline{I}\rbrace $ is homeomorphic to $S_n^+$.

LA - eng

KW - family of sets in general position; convexly connected sets; Fan-Glicksberg-Kakutani fixed point theorem; combinatorial geometry; convexly connected sets; family of sets in general position

UR - http://eudml.org/doc/248068

ER -

## References

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- Valentine F.A., Konvexe Mengen, Manheim, 1968. Zbl0157.52501MR0226495

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