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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.

Specifically, I'm interested to find if there exists some upper bound of the form $$ \dfrac{P(d)}{A_{d-1}}, $$

where $A_{d-1}$ is the surface area of the $(d-1)$-dimensional unit sphere and $P(d)$ is some polynomial of $d$ (of fixed degree). Intuitively, I want to find out if the maximum value value of the pdf of a projected log-concave distribution on the surface of the unit sphere is at most by a polynomial in $d$ larger than the value of the uniform pdf on the surface of the unit sphere.

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.

Specifically, I'm interested to find if there exists some upper bound of the form $$ \dfrac{P(d)}{A_{d-1}}, $$

where $A_{d-1}$ is the surface area of the $(d-1)$-dimensional unit sphere and $P(d)$ is some polynomial of $d$ (of fixed degree). Intuitively, I want to find out if the maximum value value of the pdf of a projected log-concave distribution on the surface of the unit sphere is at most by a polynomial in $d$ larger than the value of the uniform pdf on the surface of the unit sphere.

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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entechnic
entechnic
  • 141
  • 3

Projection of log-concave distribution on unit sphere surface

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.

Specifically, I'm interested to find if there exists some upper bound of the form $$ \dfrac{P(d)}{A_{d-1}}, $$

where $A_{d-1}$ is the surface area of the $(d-1)$-dimensional unit sphere and $P(d)$ is some polynomial of $d$ (of fixed degree). Intuitively, I want to find out if the maximum value value of the pdf of a projected log-concave distribution on the surface of the unit sphere is at most by a polynomial in $d$ larger than the value of the uniform pdf on the surface of the unit sphere.