Isoperimetric regions in spherical cones and Yamabe constants of $M\times S^1$

Given closed Riemannian manifold $(M^n,g)$ of positive Ricci curvature $Ricci(g)\ge (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M\times \mathbf{R},g+d{t}^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\times S^1$.