Generalizations of the Convex Hull I am aware of generalizations of the convex kernel, via the addition of more polygonal line segments between points in a set.  However, I wonder if there are similar generalizations for the convex hull.  If they exist, what papers exist on them, and in essence, what are they?
 A: Perhaps the best all around source for generalizations of convexity is Marcel van de Vel's book Theory of Convex Structures. He also wrote a lot of papers on this subject.
Two specific generalizations come to mind. The first is a closure system. The structure $\langle X, \mathcal{C} \rangle$ is a closure system if and only if the set $\mathcal{C}$ is a collection of subsets of $X$ that satisfies two conditions:


*

*We have $X \in \mathcal{C}$.

*For all $\mathcal{A} \subseteq \mathcal{C}$ we have $\cap \mathcal{A} \in \mathcal{C}$.


Sometime they require $\varnothing \in \mathcal{C}$.
We can define a closure operator for any subset $A$ of $X$ by the formula $$\mathsf{cl}(A) = \cap \{ C \in \mathcal{C} \colon A \subseteq C \} .$$
A convex structure satisfies an additional condition:


*

*For all $A \subseteq X$ we have $\mathsf{cl}(A) = \cup \{ \mathsf{cl} F \colon F \text{ is a finite subset of } A \} $.


In this situation the closure operator is often called a convex hull operator.
The trick is to do what is done in vector spaces without any mention of numbers. As an exercise construct the usual topology for ${\mathbb{R}}^{2}$ using just convex sets.
