  We introduce the notions of strong asymptotic uniform smoothness and
convexity. We show that the injective tensor product of strongly asymptotically
uniformly smooth spaces is asymptotically uniformly smooth. This applies in
particular to uniformly smooth spaces admitting a monotone FDD, extending a
result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our
techniques also provide a characterisation of Orlicz functions $M, N$ such that
the space of compact operators $\mathcal K(h_M,h_N)$ is asymptotically
uniformly smooth. Finally we show that $\mathcal K(X, Y)$ is not strictly
convex whenever $X$ and $Y$ are at least two-dimensional, which extends a
result by Dilworth and Kutzarova.
