What is the geometric meaning or interpretation of spaces that possess the uniform weak* Kadec-Klee property? I am writing the last part of my undergraduate thesis and I would like to add a comment in which I explain the geometric idea or, in it's absence, the inspiration to define the concept mencioned, as well as some examples of spaces that fulfill the property and examples of spaces that do not comply. I would be very grateful if you could help me with any of those things.

The definition of the topological dual space of a Banach space having the uniform weak* Kadec-Klee property is that for all ε>0, there exists δ>0 such that if {fₙ*}ₙ is a sequence in the ball of the dual that weakly converges to f* and the separation constant, defined as the infimum of the distances between any two different elements of the sequence is greater than ε, so ||f*||<1-δ. As you can see, this definition Is a little bit complicated to be geometrically understood, at least I find it so.