A $k$-simplex is defined as the convex hull of $k+1$ affinely independent vectors (vertices). A face of a simplex is a convex hull of a subset of the vertices.
A triangulation of a simplex $X$ is a collection of simplices such that the union of them is $X$ and the intersection of any two of them is a face of each of the two simplices.
For a triangulation of $\sigma_k:=\{(x_1,...,x_{k+1}):x_1+\cdots+x_{k+1}=1, x_i\ge 0\}$ and a non-boundary $(k-1)$-simplex $A$ in the triangulation, i.e., $A$ is not contained in any boundary of $\sigma_k$, I am wondering how many $k$-simplices in the triangulation does $A$ lie in?
And is there any good reference to advance understanding of the simplices?
I think the answer is 2 but I don't know how to prove it rigorous: consider the hyperplane through the vertices $v_1,...,v_k$ of $A$ in $\mathbb{R}^k$. Then some (non-zero many) vertices of the triangulation should be on one side and some (non-zero many) on the other side, as $A$ is not on the boundary of $\sigma_k$. Since the simplices in a triangulation only intersect at faces, there should be only two $k$-simplices sit on $A$, otherwise two of them sit on the same side of the hyperplane and their intersection is larger than $A$ that is in the hyperplane.