  We provide a twofold extension of Landau--Pollak uncertainty relations for
mixed quantum states and for positive operator-valued measures, by recourse to
geometric considerations. The generalization is based on metrics between pure
states, having the form of a function of the square of the inner product
between the states. The triangle inequality satisfied by such metrics plays a
crucial role in our derivation. The usual Landau--Pollak inequality is thus a
particular case (derived from Wootters metric) of the family of inequalities
obtained, and, moreover, we show that it is the most restrictive relation
within the family.
