Show that the intersection of 2 convex sets is convex but that the union of convex sets doesn't have to be?

Question

Show that the intersection of 2 convex sets is convex but that the union of convex sets doesn't have to be?

Answer 1

Suppose that there exist two points A and B in the intersection, and a point C on the line segment AB which is does not lie within the intersection. Then, the point C does not lie within both sets - that is, at least one of the two sets is lacking the point C. But, since A and B are in the intersection, they are both within that set, which contradicts the convexity of that set. This logical contradiction forces us to conclude that the original assumption - that the intersection is nonconvex - is false.

As for the second part, all you have to do is to construct two convex sets which have a nonconvex union. You could for example choose the sets [0,1]×[0,1] and [1,2]×[1,2]. Then, the line joining the points [0,1] and [1,2] passes through the point [0.5,1.5], which is not a member of the union.