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

    ***Duality map in strictly convex Banach spaces***

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