  The suspicion that the existence of a minimal uncertainty in position
measurements violates Lorentz invariance seems unfounded. It is shown that the
existence of such a nonzero minimal uncertainty in position is not only
consistent with Lorentz invariance, but that the latter also fixes the algebra
between position and momentum which gives rise to this minimal uncertainty. We
also investigate how this algebra affects the underlying quantum mechanical
structure, and why, at the Planck scale, space can no longer be considered
homogeneous.
