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I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.

Could you please provide me some references related to measuring how far a probability distribution is from the uniform one, in a convex subspace of $\mathbb{R}^d$?

More specifically, I am trying to find a distance (or, more in general, a measure of how one probability distribution is different from a second, reference probability distribution - e.g., similar to KL divergence) to exclude (or detect) distributions of a set of points that can be (informally) described as follows: Drawing $n$ points from the considered distribution, it is very likely that we have one point is very far from all the others $n-1$; considering only the others $n-1$, then it is very likely that we still have one point very far from all the others $n-2$, and so on. I was referred to as "far from being uniform" because, at the moment, it seems to be the simplest way to define a class that excludes the above-described kind of probability distributions. I hope it sounds less vague now.

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    $\begingroup$ Can you narrow this down? Otherwise you might just get a long list of distances you already know about, or pointers to generic references. "Convex subspace of $\mathbb{R}^d$" doesn't help much because nearly every general notion of distance can be applied in this setting. What have you already considered, and what do you need that they don't give you? $\endgroup$
    – Nate Eldredge
    Feb 13, 2021 at 18:20
  • $\begingroup$ Thank you @NateEldredge for the suggestion. I tried to add some specific details. $\endgroup$
    – Penelope Benenati
    Feb 13, 2021 at 18:35

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