| Title:
|
Separation of $(n+1)$-families of sets in general position in $\bold R^n$ (English) |
| Author:
|
Balaj, Mircea |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
38 |
| Issue:
|
4 |
| Year:
|
1997 |
| Pages:
|
743-748 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $S_n^+$. (English) |
| Keyword:
|
family of sets in general position |
| Keyword:
|
convexly connected sets |
| Keyword:
|
Fan-Glicksberg-Kakutani fixed point theorem |
| MSC:
|
47H10 |
| MSC:
|
52A37 |
| idZBL:
|
Zbl 0946.52002 |
| idMR:
|
MR1603706 |
| . |
| Date available:
|
2009-01-08T18:37:39Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118969 |
| . |
| Reference:
|
[1] Balaj M.: $(n{+}1)$-families of sets in general position.Beitrage zur Algebra und Geometrie 37 (1996), 67-74. Zbl 0856.52007, MR 1407806 |
| Reference:
|
[2] Fan K.: Fixed-point and minimax theorems in locally convex topological linear spaces.Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121-126. Zbl 0047.35103, MR 0047317 |
| Reference:
|
[3] Gaal S.A.: Point Set Topology.Academic Press, New York and London, 1964. Zbl 0124.15401, MR 0171253 |
| Reference:
|
[4] Glicksberg I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points.Proc. Amer. Math. Soc. 3 (1952), 170-174. Zbl 0163.38301, MR 0046638 |
| Reference:
|
[5] Hanner O., Radström H.: A generalization of a theorem of Fenchel.Proc. Amer. Math. Soc. 2 (1951), 589-593. MR 0044142 |
| Reference:
|
[6] Singer I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (in Romanian).Edit. Academiei Române, Bucureşti, 1967. MR 0235368 |
| Reference:
|
[7] Valentine F.A.: The dual cone and Helly type theorems.in: Convexity, V.L. Klee ed., Proc. Sympos. Pure Math. 7, Amer. Math. Soc., 1963, pp.473-493. Zbl 0138.43204, MR 0157285 |
| Reference:
|
[8] Valentine F.A.: Konvexe Mengen.Manheim, 1968. Zbl 0157.52501, MR 0226495 |
| . |
