I am trying to code up an algorithm I came up with, but I have the following problem with vectors whose components are a convex combination. The goal is to partition $\mathbb{R}^n$ in the following way:
For now, let $n = 2$, the convex vectors in $\mathbb{R}^2$ map the line segment from the points $(0,1)$ to $(1,0)$. The orthogonal complement of this simplex is defined by the vector $(1,1)$ which is orthogonal to the line segment. Therefore, we can also define the orthogonal complement of the line segment as the region $R_1 = \{(x_1, x_2) \in \mathbb{R}^2:\ x_2-x_1 \leq 1 \text{ and } x_2-x_1 \geq -1\}$ These two boundaries partition $\mathbb{R}^2$ into the regions whose points satisfy just one of these constraints or both constraints as shown in the picture.
In the plot, the red segment is $x_2 = 1 - x_1$ which defines all the points where $x_1 + x_2 = 1: x_1, x_2 \geq 0$, and the blue and green lines represent the functions $x_2 = 1 + x_1$ and $x_2 = -1 + x_1$ respectively. We can see that the green and blue lines partition $\mathbb{R}^2$
Now, let $n = 3$. By the same logic as above, we can find the vector $(1,1,1)$ to be orthogonal to the plane defined by $W = \{(x_1, x_2, x_3) \in \mathbb{R}^3:x_1 + x_2+x_3=1\}$. Now, I am having trouble finding the specific boundaries. What I was thinking was to first work on a triangle on the $(x_1,x_2)$-plane and find the points $(x_1,x_2,x_3)$ where the projection $(x_1,x_2,x_3) \mapsto (x_1,x_2)$ is inside this triangle, and then find a linear function that transforms the triangle in this plane to the triangle created by the convex combination in $\mathbb{R}^3$ shown below.
I tried to work on the following equilateral triangle on the $(x_1, x_2)$ plane. The triangle has a side length of $\sqrt{2}$
We can see that for $(x_1, x_2)$ to be inside the triangle in $2$ dimensions, the following constraints need to be met:
$$\text{I: } \ \ x_2 \leq \sqrt{3} x_1 + \sqrt{3/2}$$ $$\text{II: } \ \ x_2 \leq -\sqrt{3} x_1 + \sqrt{3/2}$$ $$\text{III: } \ \ x_1 \geq 0$$
which are represented by the black (I), red (II) and green (III) lines
These lines partition $\mathbb{R}^3$ into $7$ regions, where each region satisfies at least $1$ constraint, as seen in the next figure.
I have been unable to find the linear transformation that transforms this triangle as displayed in the diagram into the one that is on the first quadrant and is perpendicular to $(1,1,1)$ (with the right orientation to match the convexity requirements)
After this, I want to do this in $4$-dimensions (and stop there, since the number of constraints seems to be $2^{n+1} - 1$ and it will be very annoying), but I have resorted to being able to actually visualize the problem to find the constraints, so I have no idea how I will do this in $4D$. Is there a method that is "more algebraic" instead of relying so much on visualization?
While the general case for $n$ seems very interesting, I am more interested in $n \leq 4$.
I can see that the number of regions depends on the number and dimension of the $k$-faces of the $(n-1)$-simplex, but I am lost.
Thank you! :)