  The idea of minimum distance, familiar from R <-> 1/R duality when the string
target space is a circle, is analyzed for less trivial geometries. The
particular geometry studied is that of a blown-up quotient singularity within a
Calabi-Yau space and mirror symmetry is used to perform the analysis. It is
found that zero distances can appear but that in many cases this requires other
distances within the same target space to be infinite. In other cases zero
distances can occur without compensating infinite distances.
