Let $V$ denote a vector space. Then the following concepts make sense:
- affine subset of $V$
- affine closure (affine "hull") of a subset of $V$
Suppose $V$ is in fact a real vector space. Then the following concepts also make sense:
- convex subset of $V$
- convex closure (convex "hull") of a subset of $V$.
Now given a subset $A \subseteq V$, write $\mathrm{aff}(A)$ for the affine closure and $\mathrm{con}(A)$ for the convex closure. Clearly, both $\mathrm{aff}$ and $\mathrm{con}$ are finitary closure operators. Furthermore, we have $\mathrm{aff}(A) \supseteq \mathrm{con}(A)$. This implies that $\mathrm{con}(\mathrm{aff}(A)) = \mathrm{aff}(A)$, or in other words that every affine subset is convex. However, these are very superficial relationships between $\mathrm{aff}$ and $\mathrm{con}$.
Question 0. Does anyone know of an article or book that goes beyond the observation that $\mathrm{aff}(A) \supseteq \mathrm{con}(A)$ to actually list all the important elementary relationships between these two closure operators?
Intuitively, it seems that if $A$ and $B$ are convex subsets of $V$, then if their intersection is sufficiently "large" or "non-trivial", then $\mathrm{aff}(A) = \mathrm{aff}(B)$. For example, suppose two line segments intersect at a point. Well this is quite trivial, so we cannot expect their affine closures to agree. On the other hand, suppose the intersection of two line segments is a non-trivial line segment; in particular assume their intersection has cardinality greater than or equal to $2$. Then their affine closures should agree.
Question 1. How can we formalize the intuition that if two convex sets have a sufficiently "non-trivial" intersection, then their affine closures necessarily agree?
Bonus points if you can do it without using the concepts of "basis" or "dimension."