\((n+1)\)-families of sets in general position (Q1922679)

A family \(\overline A\) of compact sets in \(\mathbb{R}^n\) is said to be in general position if any \(m\)-dimensional flat \((0 \leq m \leq n - 1)\) intersects at most \(m + 1\) members of \(\overline A\), and a set \(A\) is called convexly connected if there is no hyperplane \(H\) such that \(H \cap A = \emptyset\) and \(A\) contains points in both the open half-spaces determined by \(H\). The author characterizes \((n + 1)\)-families \(\overline A\) of convexly connected sets by showing the equivalence of the following statements: (i) \(\overline A = \{A_1, \dots, A_{n + 1}\}\) is in general position. (ii) For any hyperplane \(H\), there exist two members of \(\overline A\) which are nicely separated by a translate of \(H\) (i.e., the separating hyperplane of the two sets \(A_i, A_j\) is disjoint from \(A_i\) or from \(A_j)\). Moreover, for any proper subset \(I\) of \(\{1,2, \dots, n + 1\}\), there exists exactly one hyperplane \(H\) strictly separating the sets \(\cup \{A_i : i \in I\}\) and \(\cup \{A_j : j \in \{1, \dots, n + 1\} \backslash I\}\) and satisfying \(d(A_1,H) = \cdots = d(A_{n + 1}, H)\), with \(d\) denoting the usual distance measure.