I am not very sure if the following problem has been treated in the literature and if so, whether it always holds:

A Banach space $X$ is isomorphic to a Hilbert space if the 'norm attaining' map $F$ from $S_{X^*}$ into $S_X$ satisfying $x^*(F(x^*)) = 1$ is a bilipschitz equivalence. Here $S_Z$ denotes the unit sphere of the Banach space $Z$.