The mirror property of metric $2$-projection

The notion of a mirror selection out of metric $2$-projection is introduced (metric $2$-projection of two elements $x_1$, $x_2$ of a Banach space onto its subspace $Y$ consists of all those elements $y\in Y$, for which the length of the broken line $x_1yx_2$ is minimal). It is proved that the existence of mirror selection out of metric $2$-projection onto every subspace having a prescribed dimension or codimemsion is a characteristic property of Hilbert space. A relation between mirror selection out of metric $2$-projection and central selection out of the usual metric projection is pointed out.