The convex hull $[X]$ of a set $X \subset \mathbb R^d$ is the set of all convex combinations $$[X] = \left\{a_1 x_1 + \ldots + a_n x_n: x_i \in X, a_i \ge 0, \sum_{i=1}^n a_i=1\right\}$$ of elements of $X$. The closed convex hull $\langle X \rangle$ is the intersection of all closed half spaces that contain $X$. I wonder is there a corresponding notion of half-space for the first definition.
Is there a family $\mathcal F$ of subsets of $\mathbb R^d$ such that each $[X] = \bigcap \{C: C \in \mathcal F: X \subset C\}?$
The answer is yes. Just take $\mathcal F$ as the collection of convex sets. This is not satisfying however so let's consider some candidates.
First let $\mathcal F_1$ be the collection of open half-spaces. The condition does not hold because of the counterexample $X = \{(x,y) \in \mathbb R^2 : y < 0\} \cup \{(0,0)\}$. Then $X$ is convex but the $\mathcal F_1$-hull is easily seen to be the lower half-plane.
Two more ideas are the following:
(2) Given an open half-space $P \subset \mathbb R^d$ and corresponding closed half-space $\overline P$ we call any set $Q$ with $P \subset Q \subset \overline P$ a half-space. Let $\mathcal F_2$ be the family of half-spaces.
The counterexample no longer works for $\mathcal F_2$.
(3) For any $a \in \mathbb R^d$ we say a convex set $X$ is $a$-maximal to mean $a \notin X$ and $X$ is not properly contained in a convex subset that does not contain $a$. Let $\mathcal F_3$ be all the convex sets that are $a$-maximal wrt some $a$.
I think the elements of $\mathcal F_3$ can be described as follows. Take an open half-space tangent to $a$. Look at the boundary and take an open $(d-1)$ dimensional half spac tangent to $a$. Look in the boundary of that half-space and proceed until you run out of dimensions.
Clearly $\mathcal F_3$ is more restrictive than $\mathcal F_2$ but the counterexample still fails. Just take the intersection of $\{y < 0\} \cup \{y=0$ and $x<\epsilon\}$ and $\{y < 0\} \cup \{y=0$ and $x<-\epsilon\}$ for all $\epsilon >0$.
I wonder do either of these families induce the usual convex hull definition?