This is just for the convex hull of finitely many points in Euclidean space, without a choice of coordinates. The axiomatization of Euclidean geometry I have in mind is Tarski's axioms, for which betweenness is a primitive notion.
- Define the convex $0$-skeleton to be the finite set of points we are starting with.
- Define the convex 1-skeleton to be the set of all points which are between any two points of the convex 0-skeleton.
- Define the convex 2-skeleton to be the set of all points which are between any two points of the convex 1-skeleton.
- Define the convex $n$-skeleton to be the set of all points which are between any two points of the convex $(n-1)$-skeleton.
Then it follows (I believe) from the upper dimension axiom that these sets have to stabilize for some $n$ (i.e. the convex $m$-skeleton equals the convex $n$-skeleton for all $m \ge n$ for some $n$).
Question: Then is the convex hull is just the convex $n$-skeleton for the smallest $n$ for which the convex $m$-skeletons equal the convex $n$-skeleton for all $m \ge n$?
Is there any better coordinate-free definition which doesn't rely upon induction?
A pointer to a reference would be appreciated.
Attempt: Using coordinates, we just have that, given finitely (an integer $p$) many points , their convex hull is the set of all points which are a convex combination of the $p$ points.
Using the coordinate realization/model of Tarski's axioms, betweenness is equivalent to being a convex combination of 2 points (i.e. the point $b$ is between the points $a$ and $c$ if and only if $b$ is a convex combination of $a$ and $c$).
So trying to verify whether this inductive definition is correct comes down to verifying that any convex combination of $p$ points can be constructed inductively from convex combinations of two points at a time. I am pretty certain this follows from distributivity and associativity.
(Not that certain though or else I wouldn't be asking this question.)