Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert space?



A characterization of Hilbert spaces by norm one projections
The answer is positive if $\dim X \geq 3$. Moreover it suffices to check the property for rank $1$ projections. See characterization (18.14) in the book [1].


However the characterization fails in dimension $2$. A rank $1$ projection is of norm $1$ iff every $x$ in the range and $y$ in the kernel are Birkhoff-orthogonal, which means that
$$ \forall t \in \mathbf{R}, \, \, \, \|x\| \leq \|x+ty\| .$$
There are non-Euclidean $2$-dimensional spaces (known as "Radon planes", see [2]) where Birkhoff-orthogonality is a symmetric relation. Such spaces have the desired property.


[1] *Amir, Dan*, Characterizations of inner product spaces, Operator Theory: Advances and Applications, Vol. 20. Basel - Boston - Stuttgart: Birkhäuser Verlag. VII, 200 p. DM 72.00 (1986). [ZBL0617.46030](https://zbmath.org/?q=an:0617.46030).


[2]
*Alonso, Javier; Martini, Horst; Wu, Senlin*, [**On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces**](https://doi.org/10.1007/s00010-011-0092-z), Aequationes Math. 83, No. 1-2, 153-189 (2012). [ZBL1241.46006](https://zbmath.org/?q=an:1241.46006).


