My definition of $d$-dimensional simplex (also called $d$-simplex) is a convex hull of $d+1$ many points in $\Bbb R^d$ (or larger space) which are affinely independent. Also, a face of a simplex is a convex hull of arbitrary subset of vertices of a simplex $\sigma$.
Currently, I'm proving the triangulation of prism is actually a triangulation. More precisely, if $\sigma$ is a $d$-simplex, then we define a prism $P = \sigma\times[0,1]$. If we let $v_0,...,v_d$ be the vertices of $\sigma$ and $v'_i$ be a vertice above $v_i$, then define a simplex $\sigma_i$ by the convex hull of $\{v_0,v_1,...,v_i,v_i',...,v_d'\}$.
In this situation, I want to show that if $F_1,F_2$ are faces of some simplices $\sigma_i,\sigma_j$ respectively, then their intersection is a convex hull of vertices both contained in $F_1$ and $F_2$. For example, if $F_1$ is a convex hull of $v_0,w_1,w_2,w_3$ and $F_2$ is a convex hull of $w_2,w_3,w_4$, then $F_1\cap F_2$ is a convex hull of $w_2,w_3$.
One known is that if the set of points we consider (in my example, $v_0,w_1,w_2,w_3,w_4$) are affinely independent, then the statement is true. But in my case, the set of points may not be affinely independent. Could you help?
Edit: I'll further assume I know 1. the interiors of the simplices $\sigma_0,...,\sigma_d$ are disjoint. 2. $\bigcup_{i=0}^d\sigma_i = P$.
Edit: A nonempty collection $\Delta$ of simplies is called simplicial complex (1) if $\sigma\in\Delta$ then all the faces of $\sigma$ is also contained in $\Delta$, (2) If $\sigma_i,\sigma_j\in\Delta$ then $\sigma_i\cap\sigma_j\in\Delta$.