Euclidean Forward–Reverse Brascamp–Lieb Inequalities: Finiteness, Structure, and Extremals

Abstract

Abstract A new proof is given for the fact that centered Gaussian functions saturate the Euclidean forward–reverse Brascamp–Lieb inequalities, extending the Brascamp–Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp–Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure, and Gaussian-extremizability for the Brascamp–Lieb inequality due to Bennett, Carbery, Christ, and Tao are generalized to the setting of the forward–reverse Brascamp–Lieb inequality.