Hello,

I'm trying to write a proof which is, in my point of view, easy to figure geometrically but not so easy to write it. I hope you can help me.

Let

be a vector space and

. The line segment that starts from

and ends in

is the set

. A set

is called convex whenever

.

Prove that the intersection

of convex sets

is a convex set.

First, I supposed that if we have

, the intersection will be

for any a, b in the naturals. As

is convex, the intersection will be too.

Second, I supposed that if

then by definition

. Is this correct?

Then I don't know exactly what I do from know: Should I suppose the intersection a set with {u,v}, and so on ({u,v,w}, ...)? I don't know if I'm going through the right direction.

I appreciate if you can help me or give me an advice about how to get confident in this kind of problem.