I try to understand the proof for the following statement: Let $C$ be a convex set (in $\mathbb{R}^n$), then $aff(C) = aff(ri(C))$, where $ri(C)$ is the relative interior of C.
I have been looking into several textbooks about convex analysis, however, all these textbooks dont proof that statement but merely refer to a theorem saying that the relative interior of a convex set $C$ is nonempty if $C$ is nonempty. And I really dont understand how these two things relate to each other.
So can someone show me a proof of the above statement regarding the affine hulls and maybe comment on how it relates to the statement about the nonempty relative interior?
This is really part of the proof that $ri\ C \ne \emptyset$. There one considers affinely independent points $x_0 \dots x_d \in C$, shows that their center $\bar x := \frac1{d+1}\sum_i x_i$ is a relatively interior point of $C$.
Then also $\frac12(x_i + \bar x)$ are in $ri \ C$ and are affinely independent. One checks that the affine hull of $x_0 \dots x_d$ and of $\frac12(x_0 + \bar x) \dots \frac12(x_d + \bar x)$ are equal. The latter proof is equal to the related result for linear subspaces: if $U\subset V$ are linear subspaces and $dim\ U = dim \ V$ then $U=V$.