I'm taking an Intro to Optimization grad course and the notion of the relative interior of a set was introduced as
The relative interior of a convex set $C$, denoted $\text{ri } C$, is the interior of $C$ as a topological subspace of $\text{aff } C$: $z \in \text{ri } C$ if there's an $r > 0$ such that $(z+rB) \cap \text{aff } C \subseteq C$.
Where $B = \{x \in \mathbb R^n : |x| < 1\}$. I'm wondering why is it that the affine, and not the convex, hull that is used in the definition. Maybe I'm missing something, but it doesn't seem economical to me in the sense that the convex hull too includes the set, has an interior that overlaps with the "intended" relative interior of the original set (as in it recovers the interior of a set that was lost in higher dimensions), but is also "smaller" than the affine hull.
If it's not a matter of convention, my hunch is that it could have something to do with an affine set being parallel to a (particular) subspace of $\mathbb R^n$, so that properties of vector spaces could be used after some translation and maybe that would be nice for later theorems. Any insight is much appreciated, thank you in advance.
Given a metric space $V$ and any vector subspace $U$, you can define the "subspace topology" on $U$ and it will also be a metric space with the same metric. Since $\textrm{aff}\, C$ is a subspace, you can define a topology on it which is (in some sense) derived from the topology of $V$, and one define a notion of an open ball using the same metric (which appears in the definition of relative interior), etc.
However, since a convex hull is not guaranteed to be a vector space, it probably won't be a metric space with the same metric (probably not even a topological vector space, since it would lack additive inverses). One could technically topologize it with the subspace topology, but it won't have the same topology as the subspace topology w.r.t. $\textrm{aff}\, C$. Since the topology w.r.t. $\textrm{aff}\, C$ is more compatible with the topology of $V$, this sort of construction is far more useful in convex analysis.