Asymptotic uniform convexity of a norm 
on an infinite-dimensional Banach space 
is positivity, for every 
, of the modulus
 |
where 
consists of the closed linear subspaces
of finite codimension.
The subspace 
may depend on the center 
and on 
, but must work for every indicated direction. The definition
is abbreviated AUC. AUC renormability instead means existence of an equivalent norm with this property. Asymptotic
midpoint uniform convexity tests opposite directions together and can hold for
a norm that is not asymptotically uniformly convex (Dilworth
et al. 2016).
