  We develop a min-max theory for the area functional in the class of locally
wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a
Riemannian manifold that is allowed to have both boundary and certain types of
edges. Fix a dimension $3 \le n+1 \le 6$. As our main theorem, we prove that
every compact locally wedge-shaped manifold $M^{n+1}$ with acute wedge angles
contains a locally wedge-shaped free boundary minimal hypersurface $\Sigma^n$
which is smooth in its interior and on its faces and is $C^{2,\alpha}$ up to
and including its edge. We can also handle the case of 90 degree wedge angles
under an additional assumption.
