My question was prompted by an earlier MO by @Daniel:

    [***Duality map in strictly convex Banach spaces***](http://mathoverflow.net/questions/125765/duality-map-in-strictly-convex-banach-spaces/130226#130226)

I will even use his symbol   $\phi$   below.

Let   $B$   be an arbitrary Banach space. Let   $S := \{x\in B:\|x\|=1\}$   be its unit sphere. Let   $\Gamma := \{f\in B^*: \|f\|=1\}$    be the unit sphere in the dual space $B^*$.

**QUESTION**   Are the following two conditions on $B$ equivalent:

 1. $B$   is isometric to a Hilbert space.
 2. There exists an isometry   $\phi: \Gamma \rightarrow S$   such that   $\forall_{f\in\Gamma}\ f(\phi(f))=1$.

**?**
 
The finite-dimensional case is especially basic.

**REMARK 0**   Perhaps similar questions were asked in the past (on MO too?)--please, let me know.

**REMARK 2** The case of   $\mathbb R^2$   and its two dual but isometric norms   $L_\infty\quad L_1$   is interesting. The general question related to the one above is to describe all Banach spaces which are isometric to their dual space. Is there any beside the Hilbert spaces and   $\mathbb R^2$   with the norm(s) just mentioned above?