Let $P$ be a convex $d$-dimensional polytope. I have two questions, related to triangulations of $P$.

*Question 1:*
Let $p$ be in the interior of $P$. Can I always find a triangulation of $P$,
such that the (or some, if not unique) $d$-dim simplex $S$ that contains $p$,
has the property that $S$ intersects the boundary of $P$, such that the intersection has same dimension as the boundary of the simplex.

*The triangulation should not add any additional vertices:*
each simplex in the triangulation must have vertices in $P$.

I think this is the property of the triangulation I wish to prove:
*A triangulation of $P$ is nice if every $k$-dim simplex in the triangulation has a $k-1$-dimensional intersection with a $k$-dimensional face of $P$.*

*Question 2:* There is the notion of "pulling triangulation",
used here by Stanley, which is a special type of triangulation.
I suspect that this triangulation is nice, but I dont know how to prove this.

I suspect this property of the triangulation is very natural, and has most likely been studied before, so I seek some reference for this.

EDIT: For the first question, it is the same as follows:

We know that since $P$ is convex, every $p \in P$ can always be expressed as $$p = a_1p_1 + \dots + a_d p_d$$ for some $p_i \in P$ and $a_i>0$ and $\sum a_i=1$. Question 1 is equivalent to that we can always choose the $p_1,p_2,\dots,p_{d-1}$ to be vertices in a face of $P$.

everysimplex has good intersection with the boundary)? $\endgroup$ – fedja Apr 2 '14 at 13:46Piecewise Linear Topology, 1969, Lemma 1.4, and were used by various other researchers before me. $\endgroup$ – Richard Stanley Nov 6 '15 at 15:53