We develop an application of Almgren-Pitts min-max theory to the study of minimal hypersurfaces in dimension 3 ≤ m + 1 ≤ 7 as well as computing the k-width of the round 2-sphere for k = 1,...,8. We show that the space of minimal hypersurfaces is non-compact for an analytic metric of positive curvature and construct a min-max unstable closed geodesic in S^2 with multiplicity 2.