Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ *isometric* with $l\_2^2$ such that $x\in E$. Does this property characterize a (separable) Hilbert space?


What about the stronger property: For any $x$ and any $n\in\mathbb{N}$ there exists a $n$-dimensional subspace $E$ isometric with $l\_2^n$ such that $x\in E$?



Do these properties characterize Hilbert spaces?
For any Banach space $X$ you can consider $X\oplus l^2$, with norm $||(x,y)||:=(||x||^2+||y||^2)^\frac{1}{2}$. Then for each $x\in X$, span$(x)\oplus l^2$ is isometric to $l^2$, so $X\oplus l^2$ is covered by isometric copies of $l^2$


