How to prove that the intersection of any finite number of convex sets is a convex set?
I have no idea.
$\begingroup$
$\endgroup$
1
asked Nov 11, 2014 at 5:24
-
$\begingroup$ Well it makes sense, because it's a going to be a subset of one or more of the convex sets,making it convex..I just don't know how to provide a rigorous proof. $\endgroup$– Aggressive Sneeze.Nov 11, 2014 at 5:33
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
Let $(S_i)$ be a convex set for $i = 1,2,\ldots,n$.
For any $x,y \in \cap_{i=1}^n S_i$, $t \in [0, 1]$, we have:
For $i = 1,2,\ldots,n$, $x \in S_i$ and $y \in S_i$ implies $tx + (1-t)y \in S_i$ by convexity of $S_i$.
Hence $tx + (1-t)y \in \cap_{i=1}^nS_i$.
Therefore $\cap_{i=1}^nS_i$ is convex.
answered Nov 11, 2014 at 5:35
-
$\begingroup$ By S_i in the 3rd line, do you mean some S_i out of the S_i's from 1 to n? $\endgroup$ Nov 11, 2014 at 6:02
-
$\begingroup$ @AggressiveSneeze. No, its valid for all $S_i$ $\endgroup$– aramNov 11, 2014 at 6:34