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Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution.

Is there any known upper bound for the probability density function of $\mathbf X/\lVert\mathbf X \rVert_2$?

I did some research on the anti-concentration properties of log-concave distributions, but I could not find anything relative.

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  • $\begingroup$ Can you clarify what kind of upper bound you're interested in, or your motivation? This might help lead to a useful answer. I am not sure if the high-dimensional case is of interest to you, but if it is, then the lower-dimensional marginals of $X/\|X\|_2$ are approximately Gaussian, and this can be made quantitative using Klartag's central limit theorem combined with thin shell estimates, Theorems 1.1 and 1.4 here: link.springer.com/article/10.1007/s00222-006-0028-8 $\endgroup$
    – Dan
    Commented Jun 6, 2023 at 15:10
  • $\begingroup$ I'm mainly interested in finding a general upper bound for the high-dimensional case, as function of the dimension $d$ (not necessarily a very tight one). The motivation comes from the multivariate standard Gaussian distribution, whose projection on the unit sphere is uniform. I would like to see what happens in the more general case of log-concave distributions and if there is any relation with the Gaussian case (for instance, in terms of the ratio between the requested upper bound and the uniform value) $\endgroup$
    – entechnic
    Commented Jun 6, 2023 at 18:07

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