Referring to the paper at http://arxiv.org/abs/0910.0608. In the proof of lemma three, can any one explain why the unit sphere Sv that be founded in the end of the proof is euclidean? and also if it so, why the normed space V must be euclidean (please see the last step is saying "Thus the unit sphere of V is a euclidean sphere, and V must be eulidean)? I thanks in advance any one can explain and simplify this to me as soon as possible.
Note: I have asked this question a few hours ago but may be in a wrong way so the people here may wouldn't understand the question, so if they just re read the question again and answer it me.
Thanks for every one who will try to help me.