Is there any clue for the supremum of projective norm in an unit ball of injective tensor product space

Question

Is there any information about the supremum, that is in a d-order tensor space $\mathcal{T}=\mathcal{R}^{n_1}\times\mathcal{R}^{n_2}\times\cdots\times\mathcal{R}^{n_d}$, $x\in \mathcal{T}$ what is $$\sup\limits_{||x||_\vee\leq 1}||x||_\wedge,$$ where $\mathcal{R}$ is the real number field and $n_1,n_2,\cdots,n_d$ are integers larger than 1.

Thank you very much!!