TITLE: Simplices in convex polytopes
QUESTION [5 upvotes]: This question is a direct generalization of:
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and  constant-line matrix into permutation matrices 
Given a convex polytope $P \in \mathbb{R}^d,$ and a point $p\in P,$ how many simplices $S$ are there such that the $S$ is the convex hull of some $d+1$ vertices of $P$ and such that $p \in S?$
I assume that this is $\#P$-complete...

REPLY [5 votes]: There are two ways to state this question, and the answers differ.  
1) Let the dimension $d$ is fixed, and the input is via $n$ vertices of the polytopes.  In this case the total number of possible simplices is polynomial, and so is the counting problem. 
2) If the dimension $d$ is arbitrary, everything falls apart.  Think of a simplex with vertices $O=(0,\ldots,0)$ and $(0,\ldots,a,\ldots,0)$, where $a\in$ {$a_1,\ldots,a_m$}.  Here $n=dm+1$.  Let $z=(c,\ldots,c)$ and consider all (closed) simplices which contain $z$. Check that #simplices containing O is a variation on the #knapsack problem, several versions of which are known to be #P-complete.  I omit the details.