Gaussian extremizers for Barthe’s reverse Brascamp–Lieb inequality

Establish whether, for any Brascamp–Lieb datum (B,p) consisting of surjective linear maps B_i: R^n → H_i with ∩_{i=1}^k ker(B_i) = {0} and positive weights p_i satisfying ∑_{i=1}^k p_i dim(H_i) = n, the existence of extremizers for Barthe’s reverse Brascamp–Lieb inequality (i.e., functions f_i for which equality holds in inequality (10)) implies the existence of Gaussian extremizers for the same datum.

Background

Barthe’s reverse Brascamp–Lieb inequality (Theorem 1.7, equation (10)) provides an optimal lower bound governed by Gaussian data for integrals constrained by linear maps B_i and weights p_i. In the forward Brascamp–Lieb inequality (Theorem 1.6), Bennett, Carbery, Christ, and Tao established that the existence of extremizers implies the existence of Gaussian extremizers. For the reverse inequality, Lehec proved that if Gaussian extremizers exist, then the corresponding Brascamp–Lieb datum is equivalent to a geometric datum.

Despite these advances, the question remains unresolved whether the mere existence of any extremizers (not assumed Gaussian) for Barthe’s reverse inequality necessarily entails the existence of Gaussian extremizers. The paper suggests that iterated convolutions and renormalizations, as used in Bennett, Carbery, Christ, and Tao for the forward inequality, might be a potential approach to address this problem.