Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 37 (1996), No. 1, 67-74.
(n+1)--Families of Sets in General Position
Mircea Balaj
Abstract.
A family of sets in ${\bf R}^n$ is said to be in general
position if any $m$--flat, \hbox{$0 \le m \le n-1,$} intersects at most $m+1$
members of the family. Using the Fan--Glicksberg--Kakutani fixed point
theorem, we prove that if $\{A\sb{1},A\sb{2},\ldots, A\sb{n+1}\}$ is a
family of compact convexly connected sets then for any proper subset $I$
of $\{1,2,\ldots,n+1\},$ there exists exactly one hyperplane $H$ which
separates strictly the sets $\cup\{A\sb{i} : i\in I\}$ and
$\cup\{A\sb{j} : j\in \{1,2,\cdots,{n+1}\}\setminus I\}$ and satisfies
$d(A\sb{1},H)=d(A\sb{2},H)=\cdots=d(A\sb{n+1},H).$