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

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9018182754516602, "perplexity_flag": "middle"}