Symmetrically separated sequences in the unit sphere of a Banach space

We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset A with the property that ‖x±y‖>1 for distinct elements x,y∈A, thereby answering a question of J.M.F. Castillo. In the case where X contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set A may be chosen in a way that ‖x±y‖⩾1+ε for some ε>0 and distinct x,y∈A. Under additional structural properties of X, such as non-trivial cotype, we obtain quantitative estimates for the said ε. Certain renorming results are also presented.