Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the affine matroid $\mathcal{M}(S)$ of $S$) with vertices in $S$. In particular


$\dim W(S) =$ number of simplices= number of bases of $\mathcal{M}(S)$.


For a given simplex $\Delta$ with vertices from $S$ define the volume form
$w\_\Delta=\chi\_\Delta d Vol$, where $\chi\_\Delta$ is the characteristic
function of $\Delta$. Let $V(S)$ be the $\mathbb{R}$-linear span of $w\_\Delta$
where $\Delta$ runs over all simplices with vertices in $S$. We have a natural
projection $\Pi: W(S)\to V(S).$ Let $K(S)$ be the kernel of $\Pi.$ Our main
goal is to describe this kernel. 


First we describe some obvious elements from
$W(S)$ lying in $K(S)$. Take an arbitrary $(d+2)$-tuple $T$ of points from $S$
spanning $\mathbb{R}^d$. Then there exists a standard element of
$K(S)$ associated to $T$. Namely, there exist exactly two triangulations of the
convex hull of $T$ by the simplices with vertices in $T$. Considering the
difference of these triangulations as an element of $W(S)$ we get the required
element of $K(S)$. For example, there are two "different" types of spanning 4-tuples
of points in $\mathbb{R}^2.$ Case 1 with the convex hull which is a 4-gon and
Case 2 with the convex hull which is a triangle. In Case 1 we have a relation
that the sum of 2 triangles = the sum of two other triangles. Thus we have an
element of $K(S)$ of the form $\Delta\_1+\Delta\_2-\Delta\_3-\Delta\_4$ In Case 2
we have that the biggest triangle $\Delta\_1$ equals either the sum of 3 smaller triangles 
(and so we have an element of $K(S)$ of the form $\Delta\_1-\Delta\_2-\Delta\_3-\Delta\_4$)
or $\Delta\_1$, with one side having 3 points from $S$, equals the sum of 2 smaller triangles 
(and so we have an element of $K(S)$ of the form $\Delta\_1-\Delta\_2-\Delta\_3$)


**Conjecture**. $K(S)$ is spanned by the above standard kernel elements
coming from spanning $(d+2)$-tuples of points from $S$.


We are able to show that Conjecture holds for $S$ for which any
$(d+2)$-subset of points is spanning.


This statement sounds to us as an exercise(?) from matroid theory. In particular, we are
sure that $\dim K(S)$ is an invariant of $\mathcal{M}(S)$. 


Can you recognize Conjecture as a known statement from
matroid theory (or, even, a version of de Rham's Theorem)?



from affine matroid to measures
This conjecture is proven in theorem 7.4 of ["Incidence matrices, geometrical bases, combinatorial prebases and matroids"](http://www.sciencedirect.com/science/article/pii/S0012365X97001076) by T.V. Alekseyevskaya and I.M. Gelfand ($n=2$) and theorem 4.5 in ["Bases in Systems of Simplices and Chambers"](http://arxiv.org/abs/math/9707218) by Alekseyevskaya for higher $n$.


A previous version of this answer described the similar situation with scissor congruence groups, but I'm no longer certain that one can get a proof using such theorems anymore. It is worth mentioning here that there is a very similar [theorem of Pachner](https://mathoverflow.net/questions/35061/a-pachner-complex-for-triangulated-manifolds) which says that two PL-homeomorphic triangulated PL-manifolds are related by a sequence of "bistellar flips", which are essentially the generating relations in your conjecture. I suspect there should be a proof of your conjecture using Pachner's theorem, but I haven't given it much thought.


