Abstract: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 $\{A_1,A_2,...,A_{n+1}\}$ is a family of compact convexly connected sets in general position in ${\bold R}^n$, then for each proper subset $I$ of $\{1,2,...,n+1\}$ the set of hyperplanes separating $\cup \{A_i: i\in I\}$ and $\cup \{A_j: j\in \overline{I}\}$ is homeomorphic to $S_n^+$.
Keywords: family of sets in general position, convexly connected sets, Fan-Glicksberg-Kakutani fixed point theorem
AMS Subject Classification: Primary 52A37; Secondary 47H10