§ Permutahedron

The permutahedron over $n$ letters is a polytope which is defined as the convex hull of all permutations of the point $(1, 2, \dots, n)$. For example, the permutahedron over 3 letters is the convex hull of the points $(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)$.
Here, we show that it can be embedded in $(d-1)$ dimensionl space, and that each point $perm((1, 2, \dots n))$ is indeed a vertex of the convex hull.
An example of a situation where a point is not in the vertex of a convex hull is convexhull(1, 2, 3) = [1, 3]. Note that 2 was used to generate the convex hull, but is not a vertex since it is the convex combination of 1, 3. We wish to prove that such a situation does not happen in the case of the permutahedron.
I have not seen an elementary proof that each point that is a permutation of the original is a vertex, so I'm recording that proof out of interest.

§ Showing that all permutations of $(1, 2, \dots, n)$ are vertices

§ Core idea

start by looking at index i of the "largest value" N in the point P. Since it's the largest point, if it's a convex combination, all other points in the convex combination must have the same "largest value" at that index i. So we can now ignore the index i and get permutations of [1..N-1]
Repeat this process to prove that all indexes of things in the convex sum must be equal to P. But this is absurd, because all vertices are distinct. Hence, the only way for this to work out is if we have only one point that we are convex-summing over. That is, P cannot be written as the convex sum of two or more vertices.

§ Formal proof

Let us be working with permutations of [1, 2, ..., N]. Pick point P. Assume P is in the convex sum of some { Xi }.
Let index(N, P) be the index of number N in P. For example, if P = [2, 3, 1], then

index(2, P) = 0 (assuming 0-based indexing)
index(3, P) = 1
index(1, P) = 2

If P is the convex sum of vertices { Xi }, all of them must have that:

Xi[index(N, P)]  = N forall Xi in the convex sum

since N is the largest value that X_I can have at any index. The only way to get the maximum value by a convex sum is for all values to be that maximum value.
So, we can now ignore dimension index(N, P), since all the vertices X_i involved in the convex combination and P have index(N, P) = N. If we project out this dimension, we are left with permutation of (1..N-1).
Repeat the same process. We will need to have all X_i to have their value at index(N-1, P) to be N-1.
Keep doing this till we get that the dimensions of all points in Xi and P are equal. But all the points in { Xi } are distinct since they are permutation of [1..n]. Hence, { Xi } can contain only a single point, and this point is P.
Hence, P cannot be written as the convex combination of other points. Hence, P is a vertex.

§ Proof that convex sum of maximum must have maximum

Assume we have values {v1, v2, ... vn} where without loss of generality, v1 <= v2 <= ... <= vn. We can always relabel otherwise.
Now we want to write vn as the convex sum of {v1, v2, ... vn}. We can draw this on the number line as:

---[v1--v2--v3--...--vn]---

• A convex sum must be inside this line segment between [v1--...--vn] ( vn is to the rightmost since it's known to be the largest value in {v1...vn}).
• So, if we try to write vn as the convex sum, we must have the coefficient of vn=1 and the coefficient of all other points =0, since any other combination will "pull the convex sum away from the right (where vn sits), towards the left".