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 $C\_i$ 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 $\_$.



Interpreting Set Notation on Theorem
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](http://postimage.org/image/xovwh0ouz/).


