Let $\big( X, \lVert \cdot \rVert \big)$ be a (real or complex) normed space. Suppose that, for every two-dimensional (vector) subspace $Y$ of $X$, the norm on $Y$ (i.e. restriction of the norm of $X$ to $Y$) satisfies the parallelogram identity and thus is induced by a certain inner product on $Y$. Can we conclude from this that the norm of $X$ is also induced from an inner product on $X$?
That is, if every two-dimensional (vector) subspace of a normed space is an inner product space, then can we prove that that normed space is itself an inner product space?