  We analyze global aspects of the moduli space of K\"ahler forms for $N$=(2,2)
conformal $\sigma$-models. Using algebraic methods and mirror symmetry we study
extensions of the mathematical notion of length (as specified by a K\"ahler
structure) to conformal field theory and calculate the way in which lengths
change as the moduli fields are varied along distinguished paths in the moduli
space. We find strong evidence supporting the notion that, in the robust
setting of quantum Calabi-Yau moduli space, string theory restricts the set of
possible K\"ahler forms by enforcing ``minimal length'' scales, provided that
topology change is properly taken into account. Some lengths, however, may
shrink to zero. We also compare stringy geometry to classical general
relativity in this context.
