Cheeger inequality for measures Given a probability measure $\mu$ on $\mathbb{R}^n$, its Poincare constant is the least number $C$ such that:
$$
\int f^2 d\mu \leq C\int \|\nabla f\|^2 d\mu
$$
for all zero mean function $f$.
Is there a version of Cheeger's inequality that works in this setting? I would assume so but was only able to find references for closed Riemannian manifolds and graphs. 
 A: Yes, the Cheeger's inequality is even known  in the general framework of a Dirichlet space $(X,d,\mathcal{E},\mu)$, where $\mu$ is a probability measure. Indeed, assume  that $\mathcal{E}$ is strictly local with a carre du champ $\Gamma$,  that Lipschitz functions are in the domain of $\mathcal{E}$ and that $\sqrt{\Gamma(f)}$ is an upper gradient in the sense that for any Lipchitz function $f$,
$
\sqrt{\Gamma(f)}(x) =\lim \sup_{d(x,y)\to 0} \frac{ |f(x)-f(y)|}{d(x,y)}.
$
In that case, if $A$ is a closed set of $X$, one defines its Minkowski exterior boundary measure by
$
\mu_+(A)=\lim \inf_{\varepsilon \to 0} \frac{1}{\varepsilon} \left( \mu(A_\varepsilon) -\mu(A) \right),
$
where $A_\varepsilon=\{ x \in X, d(x,A) <\varepsilon \}$. We can then define the Cheeger's constant of $(X,d,\mathcal{E})$ by
$
h=\inf \frac{\mu_+(E)}{\mu(E)}
$
where the infimum runs over all closed sets $E$ such that $\mu(E)\le \frac{1}{2}$. Then, according to Theorem 8.5.2 in the book by Bakry-Gentil-Ledoux , one has
$
\lambda_1 \ge \frac{h^2}{4},
$
where $\lambda_1$ is the spectral gap of $\mathcal{E}$. Note that the Buser's inequality for the Cheeger constant is also known in a very general framework (see Theorem 3.6 in this paper)
