A Note on an Asymptotic Property of the Convexity of Norms

We observe that the recently introduced property local asymptotically midpoint uniform convexity (LAMUC}) is isomorphically distinct from AMUC. Specifically, we show that $$c_0$$ or $$T^*$$, with any locally uniformly rotund norm are strictly locally AMUC. We introduce local AUC in the same spirit and give an example of locally AMUC, not locally AUC and not isomorphic to AMUC space. Following the proof in [10] we observe that these local properties are stable under finite sums with outer space having 1-unconditional basis and uniformly monotone norm.