We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the  existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension. The talk is based on joint work with E. Lieb and J. Sabin.