Characterizing affine subspaces order-theoretically Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ in a purely order-theoretic manner. Something like so:

Definition. Let $P$ denote a poset satisfying [blah]. Then $x \in P$ is called affine iff...
Proposition. The affine elements of $\mathrm{Con}(V)$ are precisely the affine subspaces of $V$.

What definition(s) are available to do this?

Clearly, it suffices to define lines in a order-theoretically way, as $X$ is affine if and only if it contains (it is bigger than) all the lines through its points.
All sets are supposed convex. The order relation is "$X$ is smaller thatn $Y$ iff $X\subset Y$"
Definition: X is polygonal if $\exists$ a convex  $Y\supset X$ such that $Y\setminus X$ is convex. P
(In practice $X$ is polygonal if and only if it is convex but not stricly convex)
Definition: $Y$ is long if it is polygonal and for any $Y\supset X$ such that $Y\setminus X$ is convex, we have $Y\setminus X$ is unbounded.
Definition: a line is a minimal long set, more precisely, $Y$ is a line if and only if $Y$ is long and has no long subset.
Fact: lines with this definition conincide with affine lines.
Proof: Any convex subset of a line is a segment or an half line. These are not long with respect to the present definition. On the other hand, let $Y$ be long and of dimension $>1$, and let $\pi$ be the affine plane containing $Y$. If $Y=\pi$ then it is not minimal, as it contains a line as a long subset. If $Y\neq \pi$ then it has a boundary and, by pushing that boundary inside $Y$, we find a convex subset of $Y$ wich is long.
Definition: $X$ is affine if and only if it is a single point or, for any line $L$ we have
$$X\cap L\neq \emptyset  \text{ in at least two points } \Rightarrow L\subset X$$
Fact: $X$ is affine iff it is an affine subspace of $V$. 
So, in general, following this scheme, you need:
1) a poset $(P,<)$ such that
i) ther is a maximal element $V$
ii) for any $Y\in P$ there is an involution map on $\{X\ :\ X<Y\}$ wich reverses the order: $X\mapsto Y\setminus X$.
2) a class of convex elements of $P$. 
3) a class of bounded (convex) elements of $P$. 
Now, points are defined as minimal elements of $P$, and $X\cap Y\neq \emptyset$ reads: "there is a point $q$ so that $q<X$ and $q<Y$".
Next, you can give the definitions of polygonal and long as above and define lines as minimal long elements. 
Finally, you have the definition of affine as an element $X$ which is either a point or so that if $X\cap L\neq\emptyset$ in two points for a line $L$, then $L<X$.
