  Motivated by the quest for an analogue of the Gromov-Hausdorff distance in
noncommutative geometry which is well-behaved with respect to C*-algebraic
structures, we propose a complete metric on the class of Leibniz quantum
compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric
resolves several important issues raised by recent research in noncommutative
metric geometry: it makes *-isomorphism a necessary condition for distance
zero, it is well-adapted to Leibniz seminorms, and --- very importantly --- is
complete, unlike the quantum propinquity which we introduced earlier. Thus our
new metric provides a natural tool for noncommutative metric geometry, designed
to allow for the generalizations of techniques from metric geometry to
C*-algebra theory.
