  The goal of the paper is to sharpen and generalise bounds involving the
Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of
the Laplacian.
  A celebrated lower bound of $\lambda_{1}$ in terms of $h$, $\lambda_{1}\geq
h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An
upper bound on $\lambda_{1}$ in terms of $h$ was established by Buser in 1982
(with dimensional constants) and improved (to a dimension-free estimate) by
Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded
below.
  The goal of the paper is two fold. First: we sharpen the inequalities
obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality
for spaces with (Bakry-\'Emery weighted) Ricci curvature bounded below by $K\in
{\mathbb R}$ (the inequality is sharp for $K>0$ as equality is obtained on the
Gaussian space). Second: all of our results hold in the higher generality of
(possibly non-smooth) metric measure spaces with Ricci curvature bounded below
in synthetic sense, the so-called $ \mathsf{RCD}(K,\infty)$ spaces.
