Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as $E(v)=$ { $x \in Vert(S)$ | $xv$ is an edge of $CH(S)$ }.

My question is best stated using an example in 2-D:

In this example, $S = ${ $v_1, v_2, v_3, v_4, v_5 $} and $v_1, v_2, v_3, v_4, v_5$ in clockwise order constitutes $Vert(S)$. There exists a line $L$ between $v_1$ and $CH(S - ${ $v_1$}$)$. If we link $v_1$ with every other 4 points, we will obtain 4 new points $p_2, p_3, p_4, p_5$ on $L$, where $p_i$ is the intersection of $L$ and the line $v_1v_i$. **You will find that { $p_2, p_5$ } $= Vert($ { $p_2, p_3, p_4, p_5$ } $)$**.

Here comes my problem. Given a finite set $S$ in higher dimensional (i.e. >= 3) vector space, we can find $Vert(S) = $ { $v_1, v_2, \cdots, v_h$ }. Now consider a point $v \in Vert(S)$. There exists a hyperplane between $v$ and $CH(S - ${$v$}$)$. If we link $v$ with every other points in $Vert(S)$, we will get a set $P$ containing $h - 1$ points on the hyperplane.

Will the points in $P$ corresponding to $E(v)$ constitute $Vert(P)$? Or is there some related theorems?

vertexandedgemake sense. $\endgroup$