I've come up with the following conjecture about convexifiability being determined by "important" sets in Banach spaces. To me, the conjecture looks quite innocuous and intuitive, but I'm having trouble getting any traction.
Conjecture: Let $X$ be a Banach space and let $K\subset B_X$ be a balanced set such that the closed, convex hull $\overline{\text{co}}(K)$ of $K$ is $B_X$. Then there exist $a>1>b>0$ such that for any $\sigma,\tau\in (0,1)$, the following are equivalent:
- For any net $(x_\lambda)\subset K$ which is weakly convergent to some $x\in B_X$ and such that $\inf_\lambda \|x_\lambda-x\|\geq \sigma$, it follows that $\|x\|\leqslant 1-\tau$.
- For any net $(x_\lambda)\subset B_X$ which is weakly convergent to some $x\in B_X$ and such that $\inf_\lambda \|x_\lambda-x\|\geqslant a\sigma$, it follows that $\|x\|\leqslant 1-b\tau$.
The result is true if we replace $X$ with $X^*$ and replace weak convergence with weak$^*$-convergence. Therefore the result is also true in the case that $X$ is reflexive, since $X=X^{**}$ and the weak and weak$^*$ topologies are the same in this case. But I can't see how to prove it without some compactness.
The difficulty seems to be that if we have a net of convex combinations such that each vector in the set is basically $x$ plus a "far out" piece in the weak topology, it is not necessarily decomposable into a convex combination of pieces each of which also have this property.
