Explicit constant for Carbery–Wright inequality

The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. However, I cannot find any reference where an explicit estimate on the constant B on the r.h.s. of the inequality is given. Knowing this constant is crucial for the application I have in mind. Are there known estimates on it? I should also say that the polynomial I have in mind is of the form $$p(g_1,g_2,\ldots,g_k)=\langle g_1\otimes g_2 \ldots\otimes g_k,A g_1\otimes g_2 \ldots\otimes g_k\rangle$$, where $$A\in\mathbb{R}^{d^k\times d^k}$$, $$\textrm{tr}(A)=0$$, and the $$g_i\in \mathbb{R}^d$$ are random vectors with i.i.d. gaussian entries. Maybe this special structure helps to obtain better anti-concentration estimates. Any help would be appreciated!

1 Answer

Have you looked at the original paper by A.Carbery and J.Wright ? Theorem 8 page 244 is the famous inequality with a sharp constant.

• what do you mean by sharp constant? I do not see an explicit constant in that Theorem. You mean this article: pdfs.semanticscholar.org/e997/… ,right? It only says "There exists an absolute constant C". Do we know what that constant is? – user134977 May 26 '20 at 12:09
• The way it is stated suggests that C is independent of the polynomial. The constant C is numerical. – user69642 May 26 '20 at 12:22
• Oh, now I understand the issue. Sure, the constant C is numerical, but my application requires bounding this constant. Note that the statement of the inequality only becomes nontrivial for $\alpha\leq C^{-2d}$, where $d$ is the degree of the polynomial. In the application I have in mind, the degree of the polynomial grows, so the inequality is nontrivial only for exponentially small $\alpha$, with the exponent depending on $C$. That is why I actually want to know $C$ in the inequality. – user134977 May 27 '20 at 13:33
• Then, I guess you should be able to verify its order of magnitude numerically by testing different polynomials in dimension 1 for the Gaussian since it does not depend on the dimension, neither on the degree, neither on the polynomial neither on the measure... – user69642 May 27 '20 at 17:45
• Ok! I thought you were interested in a non-trivial lower bound. – user69642 May 29 '20 at 8:10