“Suns” and geometric properties of the unit sphere in a Banach space

In a Banach space, even such simple objects as two-point sets have approximative properties greatly depending on the geometric structure of its unit sphere. A characterization of spcices (abstract and some concrete – $C(Q)$, $G(Q)*$, $L^1(\mu)$ and $L^{\infty}(\mu))$ is given in which every two-point set is an $\alpha$-sun (or a $\gamma$-sun).