  While studying the existence of closed geodesics and minimal hypersurfaces in
compact manifolds, the concept of width was introduced in different contexts.
Generally, the width is realized by the energy of the closed geodesics or the
volume of minimal hypersurfaces, which are found by the Minimax argument.
Recently, Marques and Neves used the $p$-width to prove the existence of
infinite many minimal hypersurfaces in compact manifolds with positive Ricci
curvature. However, whether the $p$-width can be realized as the volume of
minimal hypersurfaces is not known yet. We introduced the concept of the $(p,
m)$-width which can be viewed as the stratification of the $p$-width, and
proved that the $(p, m)$-width can be realized as the volume of minimal
hypersurfaces with multiplicities.
