Stability conjecture for the Brascamp–Lieb and reverse Brascamp–Lieb inequalities

Let $f$ be an even probability density on $\mathbb{R}$ with variance $1$, and let

$$g(t)=\frac{1}{\sqrt{2\pi}}e^{-t^2/2}$$

be the standard normal density. Let $\mu$ be an even isotropic measure on $S^{n-1}$ supported at $u_1,\ldots,u_k\in S^{n-1}$, with $\mu(\{u_i\})=c_i$. Let $\nu_n$ denote the relevant extremal isotropic measure and let $\delta_{\rm WO}(\mu,\nu_n)$ be the distance used in the conjecture. Brascamp–Lieb stability conjecture. There exist an absolute constant $\alpha>0$ and a constant $\gamma>0$ depending on $n$ such that

$$\int_{\mathbb{R}^n}\prod_{i=1}^k f(\langle x,u_i\rangle)^{c_i}\,dx\leq\exp\left(-\gamma\min\{1,\|f-g\|_1\}^{\alpha}\delta_{\rm WO}(\mu,\nu_n)^{\alpha}\right),$$ $$\int_{\mathbb{R}^n}^{\ast}\sup_{x=\sum_{i=1}^k c_i\theta_i u_i}\prod_{i=1}^k f(\theta_i)^{c_i}\,dx\geq\exp\left(\gamma\min\{1,\|f-g\|_1\}^{\alpha}\delta_{\rm WO}(\mu,\nu_n)^{\alpha}\right).$$

The conjecture seeks quantitative stability for both the Brascamp–Lieb inequality and its reverse form. The authors state that no general method was known for such a stability result, and the conjecture is presented as a proposal extending the stability estimates proved for the particular functions considered in the paper.