The following question is relatively straightforward and almost looks like an exercise from a textbook but I have no idea how to handle it. The problem is related to spaces with asymptotically uniformly convex renormings. I won't include the definition here.
Let $B_X$ be the ball of Banach space $X$. Show there exist constants $0<b<1<a$ such that the following conditions are equivalent for any Banach space $X$, any $K\subset B_X$ such that $\overline{\text{co}}(K)=B_X$ ($\overline{\text{co}}(K)$ is the closure of the convex hull of $K$), and any $0< \tau,\sigma<1$:
Whenever $(x_\lambda)\subset B_X$ is a net which is weakly convergent to some $x\in X$ and $\inf_\lambda \|x_\lambda-x\| \geqslant \tau$, it follows that $\|x\|\leqslant 1-\sigma$.
Whenever $(x_\lambda)\subset K$ is a net which is weakly convergent to some $x\in X$ and $\inf_\lambda \|x_\lambda-x\|\geqslant b\tau$, it follows that $\|x\|\leqslant 1- a\sigma$.
The obstruction is simple: How do you pass from a statement that is true for all $(x_\lambda)$ and $x$ in a set $K$ to a similar statement about all vectors in $B_X$?
I am interested in any ideas or references.
