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The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of$2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of$2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to$r$-matrices. Recently, Kolotilina, and independently the author, strengthenedSchur's bound for $2$-matrices. The main result of this paper extends thelatter result to $r$-matrices, thereby improving the result of Hardy,Littlewood, and Polya.  The proof is based on combinatorial concepts like $r$-partite $r$-matrix andsymmetrant of a matrix, which appear to be instrumental in the study of thespectral $p$-norm in general. Thus, another application shows that the spectral$p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix areequal whenever $p\geq r$. This result contributes to a classical area ofanalysis, initiated by Mazur and Orlicz around 1930.  Additionally, a number of bounds are given on the $p$-spectral radius and thespectral $p$-norm of $r$-matrices and $r$-graphs.

Combinatorial methods for the spectral p-norm of hypermatrices