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?

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.

• Describing the topology mentioned in the exercise does not require the use of real numbers. Proving that you get the usual topology will probably require the use of real numbers. – Jay Feb 5 '12 at 13:02