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While studying the existence of closed geodesics and minimal hypersurfaces incompact manifolds, the concept of width was introduced in different contexts.Generally, the width is realized by the energy of the closed geodesics or thevolume of minimal hypersurfaces, which are found by the Minimax argument.Recently, Marques and Neves used the $p$-width to prove the existence ofinfinite many minimal hypersurfaces in compact manifolds with positive Riccicurvature. However, whether the $p$-width can be realized as the volume ofminimal 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, andproved that the $(p, m)$-width can be realized as the volume of minimalhypersurfaces with multiplicities.

The (p;m)-width of Riemannian manifolds and its realization