Sharp Reverse H\"older property for A_\infty weights on spaces of homogeneous type

In this article we present a new proof of a sharp Reverse H"older Inequality for $A_{\mathrm{\infty }}$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving $A_{\mathrm{\infty }}$ constants: |M|{L^p(w)} \leq c (\frac{1}{p-1} [w]{A_p}[\sigma]{A\infty})^{1/p}, where $1<p<\mathrm{\infty }$, $\sigma =w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the measure $\mu$ and the geometric constant $\kappa$ of the quasimetric.