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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?

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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.

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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

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