“…These results were extended by Aubin for applications in Riemannian geometry [1,2]. These ideas lead mathematicians to employ rearrangement methods [9,27,29,30], to seek best constants [16,21,27], and to explore the role of symmetry in various functional inequalities [8,13,17,22]. In recent years, researchers have also been using new techniques such as optimal transport to pursue these types of results [3,12,23].…”