New examples of strongly subdifferentiable projective tensor products

We prove that the norm of $X{\widehat{\otimes }}_{\pi }Y$ is SSD if either $X={\mathrm{\ell }}_p(I)$ for $p>2$ and $Y$ is a finite-dimensional Banach space such that the modulus of convexity is of power type $q<p$ (e.g. if $Y^{\ast }$ is a subspace of $L_q$) or if $X=c_0(I)$ and $Y^{\ast }$ is any uniformly convex finite-dimensional Banach space. We also provide a characterisation of SSD elements of a projective tensor product which attain its projective norm in terms of a strengthening of the a local Bollobás property for bilinear mappings.