  As is well known, the smallest possible ratio between the spectral norm and
the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and
is (up to scalar scaling) attained only by matrices having pairwise orthonormal
rows. In the present paper, the smallest possible ratio between spectral and
Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also
called the best rank-one approximation ratio in the literature, is
investigated. The exact value is not known for most configurations of $n_1 \le
\dots \le n_d$. Using a natural definition of orthogonal tensors over the real
field (resp., unitary tensors over the complex field), it is shown that the
obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a
tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal
or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the
field. A connection between the (non)existence of real orthogonal tensors of
order three and the classical Hurwitz problem on composition algebras can be
established: existence of orthogonal tensors of size $\ell \times m \times n$
is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz
problem. Some implications for higher-order tensors are then given. For
instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$
do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is
more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m
\le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations
for spectral norm computation are presented.
