This question is regarding a special case of this question, for which it is plausible the details are known. The Carbery-Wright inequality is an "anti-concentration inequality" that states the following

There exists an absolute constant $C$ such that, if $X$ is a Banach space, $p : \mathbb{R}^n → X$ is a polynomial of degree at most $d$, $0<q<\infty$, and $\mu$ is a log-concave probability measure on $\mathbb{R}^n$, then

$$\left(\int \lVert p(x)\rVert^{q/d}\mu(x)\right)^{1/q}\alpha^{-1/d}\mu\{x\in\mathbb{R}^n\mid \lVert p(x)\rVert\leq \alpha\}\leq Cq.$$

This can be used to (among other things) obtain anti-concentration inequalities for the $\ell_2$ norm of log-concave random variables, i.e. for (a specific) degree 2 polynomial.

My application requires knowledge of the constant $C$ though (or some bound on it). It does not appear to be known in the general case. Is it known in this simpler case?

This question is regarding a special case of this question, for which it is plausible the details are known. The Carbery-Wright inequality is an "anti-concentration inequality" that states the following

There exists an absolute constant $C$ such that, if $X$ is a Banach space, $p : \mathbb{R}^n → X$ is a polynomial of degree at most $d$, $0<q<\infty$, and $\mu$ is a log-concave probability measure on $\mathbb{R}^n$, then

$$\left(\int \lVert p(x)\rVert^{q/d}\mu(x)\right)^{1/q}\alpha^{-1/d}\mu\{x\in\mathbb{R}^n\mid \lVert p(x)\rVert\leq \alpha\}\leq Cq.$$

This can be used to (among other things) obtain anti-concentration inequalities for the $\ell_2$ norm of log-concave random variables, i.e. for (a specific) degree 2 polynomial.

My application requires knowledge of the constant $C$ though (or some bound on it). It does not appear to be known in the general case. Is it known in this simpler case?

Edit: In the simpler case of Gaussian measures, there appears to be a result known here.