The case of equality in geometric instances of Barthe's reverse Brascamp-Lieb inequality (Q6608544)

The Geometric Brascamp-Lieb inequality has been known in the form \N\[\N\int_{{\mathbb{R}}^n}\prod_{i=1}^kf_i(P_{E_i}x)^{c_i}dx\leq\prod_{i=1}^k\bigg(\int_{E_i}f_i\bigg)^{c_i}\N\]\Nfor subspaces \(E_1,\dots, E_k\) of \({\mathbb{R}}^n\) and \(c_1, \dots, c_k>0\) satisfying the Geometric Brascamp-Lieb data relation \N\[\N\sum_{i=1}^kc_iP_{E_i}=I_n,\N\]\Nafter the works of Brascamp-Lieb, Ball, Barthe et al. The extremals \(f_i, i=1,2,\dots, k\) for which the equality of Geometric Brascamp-Lieb inequality holds were classified later by Valdimarsson. In this paper, the authors classify the extremals for which the Barthe's reverse Geometric Brascamp-Lieb inequality \N\[\N\int^*_{{\mathbb{R}}^n}\sup_{x=\sum\limits_{i=1}^kc_ix_i,\ x_i\in E_i}\prod_{i=1}^kf_i(x_i)^{c_i}dx\geq\prod_{i=1}^k\bigg(\int_{E_i}f_i\bigg)^{c_i}\N\]\Nholds with equality.\N\NFor the entire collection see [Zbl 1527.46003].