Effective Dimensions in Grothendieck--Hölder Inequalities

What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely,