Proof that the intersection of any finite number of convex sets is a convex set

Question

How to prove that the intersection of any finite number of convex sets is a convex set?

I have no idea.

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. — Aggressive Sneeze., Nov 11 '14 at 05:33

score 20 · Accepted Answer · answered Nov 11 '14 at 05:35

20

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 '14 at 05:35

Empiricist

By S_i in the 3rd line, do you mean some S_i out of the S_i's from 1 to n? – Aggressive Sneeze. Nov 11 '14 at 06:02
@AggressiveSneeze. No, its valid for all $S_i$ – aram Nov 11 '14 at 06:34

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