In fcc structure, conrer atoms do not touch each other (atoms 1 and 2), but every face centre atom touches corner. Moreover, every face centre atom touches every other face centre atom provided it is not the opposite face centre atom in an fcc unit cell.
$\mathrm{I}.$ Atoms 3 and 4 are touching each other where centre-to-centre distance $=\frac{\mathrm{a}}{\sqrt{2}}$
$\mathrm{I}\mathrm{I}.$ Atoms 1 and 2 are not touching each other
$\mathrm{I}\mathrm{I}\mathrm{I}.$ Atoms 2 and 4 are touching each other where centre-to-centre distance $=\frac{\mathrm{a}}{\sqrt{2}}$ .