The equality case in Cheeger's and Buser's inequalities on
$\mathsf{RCD}$ spaces

We prove that the sharp Buser's inequality obtained in the framework of
$\mathsf{RCD}(1,\infty)$ spaces by the first two authors 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
$\mathsf{RCD}(K,\infty)$ 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. As a
consequence of the two main results, we obtain improved versions of Buser's and
Cheeger's inequalities for $\mathsf{RCD}(K,N)$ spaces which are new even for
smooth Riemannian manifolds of dimension $N$ with Ricci curvature bounded below
by $K\in \mathbb{R}$.