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 et al. (J Math Anal Appl 402(1):297–307, 2013). Our techniques also provide a characterisation of Orlicz functions M, N such that the space of compact operators is asymptotically uniformly smooth. Finally we show that is not strictly convex whenever X and Y are at least two-dimensional, which extends a result by Dilworth and Kutzarova (Function Spaces (Edwardsville, IL, 1994), Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1995).