  In three spatial dimensions, the Compton wavelength $(R_C \propto M^{-1}$)
and Schwarzschild radius $(R_S \propto M$) are dual under the transformation $M
\rightarrow M_{P}^2/M$, where $M_{P}$ is the Planck mass. This suggests that
there is a fundamental link -- termed the Black Hole Uncertainty Principle or
Compton-Schwarzschild correspondence -- between elementary particles in the $M
< M_{P}$ regime and black holes in the $M > M_{P}$ regime. In the presence of
$n$ extra dimensions, compactified on some scale $R_E$, one expects $R_S
\propto M^{1/(1+n)}$ for $R < R_E$, which breaks this duality. However, it may
be restored in some circumstances because the effective Compton wavelength
depends on the form of the $(3+n)$-dimensional wavefunction. If this is
spherically symmetric, then one still has $R_C \propto M^{-1}$, as in the
$3$-dimensional case. The effective Planck length is then increased and the
Planck mass reduced, allowing the possibility of TeV quantum gravity and black
hole production at the LHC. However, if the wave function is pancaked in the
extra dimensions and maximally asymmetric, then $R_C \propto M^{-1/(1+n)}$, so
that the duality between $R_C$ and $R_S$ is preserved. In this case, the
effective Planck length is reduced but the Planck mass is unchanged, so TeV
quantum gravity is precluded and black holes cannot be generated in collider
experiments. Nevertheless, the extra dimensions could still have consequences
for the detectability of black hole evaporations and the enhancement of
pair-production at accelerators on scales below $R_E$. Though
phenomenologically general for higher-dimensional theories, our results are
shown to be consistent with string theory via the minimum positional
uncertainty derived from $D$-particle scattering amplitudes.
