Polya-Szego inequality and Dirichlet \(p\)-spectral gap for non-smooth spaces with Ricci curvature bounded below (Q1989422)

The first aim of this paper is to generalize the Polya-Szego and the Bérard-Meyer inequalities to non-smooth spaces with Ricci curvature bounded below in a synthetic sense. In the second part, the authors prove a new rigidity result for Polya-Szego inequality and an almost rigidity result for the Dirichlet \(p\)-spectral gap in the framework of \(\mathrm{RCD}(K, N)\) spaces, which are interesting even for smooth Riemannian manifolds with \(\mathrm{Ricci}\geq K > 0\).