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$$(n+1)$$-families of sets in general position. (English) Zbl 0856.52007
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.

##### MSC:
 52A37 Other problems of combinatorial convexity
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