If the unit sphere of a normed space is homogeneous is the space an inner product space?

Solution 1:

The answer to the bonus question in full generality is no. A Banach space is called transitive if for every unit vectors $u,v$ there is a surjective linear isometry mapping $u$ to $v$. There are nonseparable transitive Banach spaces which are not Hilbert spaces. The Banach-Mazur rotation problem asks whether every separable transitive Banach space is a Hilbert space. This remains unsolved. You can find a bunch of references by googling "transitive Banach space" or "Banach-Mazur problem". There is an MO thread with a proof for finite dimensions and a nonseparable counterexample.