# The union of two convex sets is not convex

• Jan 30th 2010, 09:55 AM
Gemini
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.)
• Jan 30th 2010, 10:33 AM
tonio
Quote:

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...(Wink) ), but you can easily build an easy example: take the two tangential circles

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

Tonio
• Jan 30th 2010, 11:02 AM
Gemini
Thanks, but if my example is correct, I'll just use it. If you don't want drawings, you better tell it ;)
• Jan 30th 2010, 01:22 PM
tonio
Quote:

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 ;)