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

Thomas A. Courtade, Jingbo Liu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)3300-3350
Number of pages51
JournalJournal of Geometric Analysis
Issue number4
StateAccepted/In press - 2020
Externally publishedYes


  • Barthe inequalities
  • Brascamp–Lieb inequalities
  • Functional inequalities

ASJC Scopus subject areas

  • Geometry and Topology


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