Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let

$$ X_f := \{x\in X : f(x)\ge 0\} $$

("*functional*" means "*non-zero linear functional*"). Also, given a topological space $E$ and its topological subspace $A$, a retraction $r:E\rightarrow A$ is defined as a continuous map such that $r(x)=x$ for every $x\in A$.

**CONJECTURE:**

Let $\,X$ be an arbitrary Banach space such that for every functional $\ f:X\rightarrow \mathbb R\ $ there is a retraction $\ r:X\rightarrow X_f\ $ such that

$$ \forall_{x\ y\ \in\ X\setminus X_f}\ \ \ |r(x)-r(y)| = |x-y| $$

Then $X$ is isometric to a Hilbert space.

REMARK:I think that questions of this type were popular in the past in the case of finite dimensional spaces (mostly 3-dim?). I am not aware of the general case (I am not a specialist thus I have to ask :-). I still believe that my usage of thelanguage here is new (even in the finite-dimensional case).retraction