For
A brief note on Freudenthal Triangulation using an example (taken from section 4.2 in Lovejoy, W. S. (1991). Computationally feasible bounds for partially observed Markov decision processes. Operations Research, 39(1), 162–175.):
Let us consider an example in $R^3$. We want to triangulate the case of three dimensions. A simplex $$ \Pi(s)= \{\pi \in 3-D is given as R^3:\pi_i \ge 0,,i=1,2,3 \sum_{i=1}^3 \pi_i=1 \}$$ We first triangulate the $R^3$ and then by some mapping induce a convex hull of triangulation in $\Pi(s)$.
To Triangulate $R^3$, consider a constant $M$ and the set of vertices (0,0,1), ($$G'=\{(q_1,q_2,q_3):M=q_1\ge q_2 \ge q_3 \ge 0\}$$. Also define the non-singular matrix $$B=\frac{1}{M} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 ) and & 0 & 1For $q \in G', Bq = \frac{1}{M} (1,0,0). An exemplary FT M-q_2,q_2-q_3,q_3), Bq \in G$ where $$G = \{\frac{m}{M}| m \in I_+^3, \sum m_i = M\}$$ where $I_+^3$ is the set of this example positive integers. The set $G$ is called the simplices set of regular grid points in $\Pi(s)$. As I said before we triangulate $R^3$ get the vertices in set $G'$ and then transform them to the vertices of set $G$. consider the following example:
The freudenthal triangulation in $R^3$ with M =2 will give us the vertices v1 (basically the set $G'$) as $(2,0,0), (2,2,0), (2,2,2), (2,1,0), (2,1,1), (2,2,1)$ . The vertex in the set $G$ are $v_1 = (0,0,1), v2 1,0,0), v_2 = (0,1,0), v3 v_3 = (1,0,0), v4 0,0,1), v_4 = (0.5,0.5,0), v5 v_5 =(0,0.5,0.5) (0.5,0,0.5)$ and v6 $v_6 = (0.5,0,0.5).
Not sure how to draw it here but on paper you will see the original simplex (which will be a triangle) divided into 4 sub-triangles 0,0.5,0.5)$. The ordering is restored with the vertices (v1,v5,v6), (v2,v4,v5),(v3,v4,v6) and (v4,v5,v6)transformation, i.e., $(2,0,0)$ refers to $1,0,0$.
I believe it might be a simple question with simple solution but I am not able to figure it out. I also think that there might be some order in which the vertices are generated and may correspond with vertices of the simplices in the partition. Can you please point me to some reference.
