I am looking for a continuous function $f$ mapping each point $(x_1,x_2,x_3)$ on the standard triangle (this means $\sum_{i}x_i=1$ and $\forall i: x_i\geq 0$) to another point $(y_1,y_2,y_3)$ on the standard triangle, with the following property.
If, for some $i$, $x_i=0$, then:
- $y_i$ is a maximum element in $(y_1,y_2,y_3)$;
- For some $j\neq i$, $y_j=0$.
I could find the value of $f$ in some special points. For example, on the face $x_3=0$:
- $f(1,0,0) = (0,1/2,1/2)$ [since both $y_2$ and $y_3$ must be maximum, and $y_1$ must be zero].
- $f(1/2,1/2,0) = (0,0,1)$; [since $y_3$ must be maximum, and some of $y_1,y_2$ must be zero].
- $f(0,1,0) = (1/2,0,1/2)$.
The function on the boundary can be intrapolated between these points, e.g.:
- $f(3/4,1/4,0) = (0,1/4,3/4)$;
- $f(1/4,3/4,0) = (1/4,0,3/4)$;
But how can I extend this to a continuous function on the entire standard simplex?