There are many, many different notations for the interior of a set. That is why you should always define the terms you post.
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.