0
$\begingroup$

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 $𝐶_𝑖$.

$\endgroup$
4
  • 7
    $\begingroup$ What is the question? $\endgroup$ Feb 15, 2013 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, 2013 at 16:14
  • $\begingroup$ For part (ii)* If the intersection.... $\endgroup$
    – dirtymike
    Feb 15, 2013 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 $i\neq j$. $\endgroup$ Feb 15, 2013 at 17:52

1 Answer 1

0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.