New examples of strongly subdifferentiable projective tensor products

We prove that the norm of $X\mathbin{\widehat{\otimes}_\pi} Y$ is SSD if either $X=\ell _p(I)$ for $p \gt 2$ and $Y$ is a finite-dimensional Banach space such that the modulus of convexity is of power type $q \lt p$ (e.g. if $Y^*$ is a subspace of $L_q$), or if $X=c_0(I)$ and $Y^*$ 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 local Bollobás property for bilinear mappings.