# Thread: The union of two convex sets is not convex

1. ## The union of two convex sets is not convex

Can I demonstrate, using Venn Diagrams, that a union of two convex sets is not necessarily convex simply by drawing something like this and then drawing a line from the top of one circle to the top of another? (The line would go outside the circles, indicating the union is not convex.)

2. Originally Posted by Gemini
Can I demonstrate, using Venn Diagrams, that a union of two convex sets is not necessarily convex simply by drawing something like this and then drawing a line from the top of one circle to the top of another? (The line would go outside the circles, indicating the union is not convex.)

Some wouldn't accept a drawing as a proof (I, for one... ), but you can easily build an easy example: take the two tangential circles

$\displaystyle (x-1)^2+y^2=1\,,\,\,(x+1)^2+y^2=1$ ,and now prove that the line joining the points $\displaystyle (1,1\slash 2)\,,\,\,(-1,1\slash 2)$ is not completely contained in the union of these two circles.

Tonio

3. Thanks, but if my example is correct, I'll just use it. If you don't want drawings, you better tell it

4. Originally Posted by Gemini
Thanks, but if my example is correct, I'll just use it. If you don't want drawings, you better tell it

Tonio

5. I just wouldn't feel right copying your example, at least not when I could come up with something of my own.

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# show that union of two convex set is not convex

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