We study edge-cone Einstein metrics (in the sense of LeBrun) on smooth closed manifolds. We first prove that, for such an edge-cone Einstein metric $h$ of cone angle $2\pi \beta\,(0 < \beta < 1)$ on a smooth closed $n$-manifold $M$ with $n \geq 3$, there exist $C^2$ Riemannian metrics $\{g_j\}_j$ such that the limit inferior of these Yamabe constants $\{Y(M, [g_j])\}_j$ is at least the Yamabe constant $Y(M, [h]) = R_h\,V_h^{2/n}$. This implies that the Yamabe invariant $Y(M)$ of the {\it smooth} manifold $M$ is estimated by the Yamabe constant $Y(M, [h])$ of the {\it singular} Einstein manifold $(M, h)$ from below. We also show an example of edge-cone Einstein manifolds of cone angle $2\pi \beta\,(\beta > 1)$ which have no Yamabe metrics.

This is a joint work with Ilaria Mondello (Cr\'eteil, FR).