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Put \\rho=\\max_{1\\leq i \\leq n-1}\\{\\rho_i\\} and\n  X_{\\rho}= { x \\in X: \\rho(dx) < \\infty for some d > 0 }. In this paper we define a new class of s-norms (norms if s=1) on X_{\\rho}. In particular, our defintion generalizes in a natural way the Orlicz-Amemiya and Luxemburg norms defined for s-convex semimodulars. 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