The equality case in Cheeger's and Buser's inequalities on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="sans-serif">RCD</mml:mi></mml:math> spaces

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