  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, strengthened
Schur's bound for $2$-matrices. The main result of this paper extends the
latter 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 and
symmetrant of a matrix, which appear to be instrumental in the study of the
spectral $p$-norm in general. Thus, another application shows that the spectral
$p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix are
equal whenever $p\geq r$. This result contributes to a classical area of
analysis, initiated by Mazur and Orlicz around 1930.
  Additionally, a number of bounds are given on the $p$-spectral radius and the
spectral $p$-norm of $r$-matrices and $r$-graphs.
