  We show that two important quantities from two disparate areas of complexity
theory --- Strassen's exponent of matrix multiplication $\omega$ and
Grothendieck's constant $K_G$ --- are intimately related. They are different
measures of size for the same underlying object --- the matrix multiplication
tensor, i.e., the $3$-tensor or bilinear operator $\mu_{l,m,n} : \mathbb{F}^{l
\times m} \times \mathbb{F}^{m \times n} \to \mathbb{F}^{l \times n}$, $(A,B)
\mapsto AB$ defined by matrix-matrix product over $\mathbb{F} = \mathbb{R}$ or
$\mathbb{C}$. It is well-known that Strassen's exponent of matrix
multiplication is the greatest lower bound on (the log of) a tensor rank of
$\mu_{l,m,n}$. We will show that Grothendieck's constant is the least upper
bound on a tensor norm of $\mu_{l,m,n}$, taken over all $l, m, n \in
\mathbb{N}$. Aside from relating the two celebrated quantities, this insight
allows us to rewrite Grothendieck's inequality as a norm inequality \[
\lVert\mu_{l,m,n}\rVert_{1,2,\infty}
=\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\lVert X\rVert_{1,2}\lVert
Y\rVert_{2,\infty}\lVert M\rVert_{\infty,1}}\le K_G. \] We prove that
Grothendieck's inequality is unique: If we generalize the $(1,2,\infty)$-norm
to arbitrary $p,q, r \in [1, \infty]$, \[
\lVert\mu_{l,m,n}\rVert_{p,q,r}=\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\|X\|_{p,q}\|Y\|_{q,r}\|M\|_{r,p}},
\] then $(p,q,r )=(1,2,\infty)$ is, up to cyclic permutations, the only choice
for which $\lVert\mu_{l,m,n}\rVert_{p,q,r}$ is uniformly bounded by a constant
independent of $l,m,n$.
