## Uniformity on a convex subspace of $\mathbb{R}^n$

Let's consider a set of points (in cartesian coordinates) in a convex subspace of $\mathbb{R}^n$ and more precisely in a subspace of the standard (n-1)-simplex ($x_i \in[0,1]$ s.t. $\displaystyle \sum_{i=1}^{n} x_i = 1$ ).

A typical subspace of mine would be the $x_i \in[a_i,b_i]$ s.t $\displaystyle \sum_{i=1}^{n} x_i = 1$ where $a_i,b_i \in(0,1)$

How can you test (or measure how) those points are uniformly distributed over this subspace ?

For the moment the only hint I had looking at articles is the distance-to-boundary test but it seems hard to evaluate the minimum distance of a given point to the simplex (or to a simplex-subspace) "boundaries" with a general formula.

The aim is to implement this test in R.

Thanks for your help.

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