In an article I am reading, they state a theorem from Victor Klee's "On Certain Intersection Properties of Convex Sets" and I am having trouble picturing this. The statement is:

Let $C$ and $C_1, ..., C_n$ be closed convex sets in a Euclidean space satisfying: (i) $C \cap \bigcap_{i \neq j}^n C_i \neq \emptyset$ for $j=1,...,n$, and (ii) $C \cap \bigcap_{i=1}^n C_i = \emptyset$. Then $C \nsubseteq \bigcup_{i=1}^n C_i$.

I want to say (i) $C$ and $C_i𝑖$ taken separately share at least some point. (ii) But, when we take the intersection of $C_i$, there are not points shared among any sets. Then, this means that $C$ is not a subset of the union of all $𝐶_𝑖$.

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    $\begingroup$ What is the question? $\endgroup$ – Mariano Suárez-Álvarez Feb 15 '13 at 6:11
  • $\begingroup$ I do not know whether or not my interpretation is correct. For part (i), is this the intersection of the sets taken separately? For part (i) If the intersection of the C_i's is taken first, then it is empty? $\endgroup$ – dirtymike Feb 15 '13 at 16:14
  • $\begingroup$ For part (ii)* If the intersection.... $\endgroup$ – dirtymike Feb 15 '13 at 16:16
  • $\begingroup$ dirtymike. Perhaps it would help to say that statement (i) is actually n statements, a separate statement for each $j$. For each j, one takes the intersection of the other $C_i$, that is, for $i\neq j$. $\endgroup$ – Joel David Hamkins Feb 15 '13 at 17:52

Your question is ambiguous, but if you mean that you can't picture a nontrivial situation satisfying the assumptions, here's a quick example.

Four convex subsets of the plane giving an example to the assumptions. $C$ is the big square and $C_1,\ldots,C_3$ are the triangles. Picture.

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