The equality case in Cheeger's and Buser's inequalities on RCD spaces

We prove that the sharp Buser's inequality obtained in the framework of RCD(1,∞) spaces by the first two authors [29] is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of RCD(K,∞) spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained.