# Condition on the point cloud matrix making the points “generic” in the uniform sense

For a matrix $$X\in\mathbb{R}^{d\times n}$$, what condition can I impose on $$X$$ to make the collection of its columns generic in the sense that they look like the result of uniformly sampling a convex region in $$\mathbb{R}^d$$? Ideas or relevant papers are welcome and appreciated. Thanks.

• So, the question is on uniformly sampling a convex region, right? – Rodrigo de Azevedo Oct 26 '19 at 5:55
• No, the question is on "what conditions I can impose on X" to make its columns look like uniform point cloud on a convex region. – Min Wu Oct 26 '19 at 5:58
• Why make it look like the original if you can get the original? – Rodrigo de Azevedo Oct 26 '19 at 6:28
• You want something like a likelihood measure, that given $X$ will tell you how "likely" it is to come from a uniform sampling on a convex? As it is, it is unclear what you are asking. Any distribution of points may come from such a sampling, although possibly with a low likelihood. – Federico Poloni Oct 26 '19 at 8:23
• Sorry for the confusions. I will restate the problem again. – Min Wu Oct 26 '19 at 16:46