Let $C$ be the unit cube in $\mathbb{R}^n$. I want to simplicize $C$ (i.e. partition it into simplexes), but I have to follow these two rules:
- I can't add any vertices to $C$. The vertices of each simplex in the partition must be $n+1$ of the $2^n$ vertices in $C$.
- A hyperplane is "positively-oriented" if its normal vector $v$ satisfies $v \ge 0$ (i.e. $v_1 \ge 0, \dots, v_n \ge 0$) or $v \le 0$. The second requirement is that every face of every simplex in the partition must be positively oriented.
Note that every face of $C$ is positively-oriented.
It's easy to do this for $\mathbb{R}^2$: divide the square with a line from its northwest corner to its southeast corner.
Can it always be done in $n$ dimensions? Or can we prove that it is sometimes impossible?