Now to question. The second is easier to prove that the first.
If then suppose that are positive numbers such that . Now such that
Now you need to show that which shows that WHY?
The first problem is a bit more involved. You must show that no point on the path from two points in contains a point of the , the boundary of (see what I mean about defining terms?)
There is a standard concept of an internal point of a convex set. At first, that is what I thought you meant. However, that does not appear to be the case here. So you most lookup that concept. Then prove the theorem that: each point of the interior of a convex subset of a topological vector space is an internal point.
The proof depends upon the continuity of scalar multiplication.
Be warned: I will not do this for you.