Asymptotic marginals of uniformly-distributed simplex

Question

this is my first post. I am studying manifolds defined by linear transformations and noticed an interesting phenomenon I need to explain and don't know where to look. If I have an $N\times D$ maxtrix ${\bf A}$, where the first column is the vector of 1's, and where $D<N$, and I transform the $N\times 1$ vector ${\bf x}$ with positive elements, I get ${\bf z}= {\bf A} {\bf x}$ where ${\bf z}$ is a $D \times 1$ vector. Now with ${\bf z}$ fixed, I look at all the positive-valued points ${\bf x}$ that map to that value ${\bf z}$, I see a manifold of dimension $N-D$ in ${\cal R}^N$. The manifold is bounded because the sum of the positive-valued elements is fixed by the first element of ${\bf z}$. For $N=3$ and $D=1$, and ${\bf z}=1$, the manifold is the triangular-shaped simplex that has the vertices [100],[010],[001]. Now here's what's interesting. If I sample uniformly on this manifold (uniform is the maximum entropy distribution) and look at the marginal distributions of the elements of ${\bf x}$, I see this converging for large $N$ to an exponential density. Therefore, samples drawn uniformly on this high-dimensional simplex has marginals that are asymptotically exponential. I have found this usesuful because I could use this property to approximate the mean (centroid) of the manifold. I was wondering if anyone knows of work that either finds the centroid of this manifold, and/or knows any existing work on the asymptotically exponential marginals. Thanks in advance!