  Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le
5$. We show that, if the metric $g$ is generic, then $M$ contains infinitely
many geometrically distinct constant mean curvature hypersurfaces, each
enclosing half the volume of $M$. As an essential part of the proof, we develop
an Almgren-Pitts type min-max theory for certain non-local functionals of the
general form $$\Omega \mapsto \operatorname{Area}(\partial \Omega) -
\int_\Omega h + f(\operatorname{Vol}(\Omega)).$$
