Let $x\_1, ..., x\_n\in\mathbb{R}^d$. We know that a point $x$ in the convex hull $\text{conv}(x\_1, ..., x\_n)$ may be expressed as convex combinations $x=\sum\_{i=1}^n r\_ix\_i=\sum\_{i=1}^n s\_ix\_i$ with distinct $(r\_1, ..., r\_n)$ and $(s\_1, ..., s\_n)$ with $r\_i, s\_i\geq 0$, for all $i$ and $\sum\_{i=1}^n r\_i=1=\sum\_{i=1}^n s\_i$. With this fact, I have the following question:  

For $x\in \text{conv}(x\_1, ..., x\_n)$, what is the number of distinct ways (i.e. distinct coefficients $(r\_1, ..., r\_n)$) that $x$ can be expressed as a convex combination of $x\_1, ..., x\_n$? Is there a simple answer/formula in terms of geometry under appropriate condition (like, in general position) on the points? Any reference of books or papers is appreciated. Thanks



Overlap count of convex combination of points
Let $F$ be the smallest face of the polytope $\text{conv}(x\_1, \dots, x\_n)$ containing $x$. If $F$ is a simplex intersecting $\{x\_1, \dots, x\_n\}$ only in its vertices, then there is a unique convex combination giving $x$: the points $x\_i$ that do not belong to $F$ must have the coefficient $r\_i$ equal to $0$. Otherwise there are continuum many such convex combinations, as user44191 mentioned.


