  We prove that if a closed unit volume Riemannian manifold, $(M^n, g)$, has
Ricci curvature bounded from below by r>0 then the Yamabe constant of the
conformal class of $g$ is at least $n.r$. This inequality has already been
proved by S. Ilias (Constantes explicites pour les inegalites de Sobolev sur
les varietes riemannienes compactes, Ann. Inst. Fourier 33, 151-165). The
equality is achieved if the metric is Einstein (with Ricci curvature r). This
implies for instance that if $h$ is the Fubini-Study metric on $CP^2$ and $g$
is any other metric on $CP^2$ with $Ricci(g) \geq Ricci(h)$ then $Vol(CP^2, g)
\leq Vol(CP^2, h)$.
