  It is generally believed that any quantum theory of gravity should have a
generic feature --- a quantum of length. We provide a physical ansatz to obtain
an effective non-local metric tensor starting from the standard metric tensor
such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order
of the Planck length $L_{P}$. This prescription leads to several remarkable
consequences. In particular, the Euclidean volume $V_D(\ell,\ell_{0})$ in a
$D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell
_{0}) \propto \ell_{0}^{D-2} \ell^2$ when $\ell \sim \ell_{0}$, while it
reduces to the standard result $V_D(\ell,\ell_{0}) \propto \ell^D$ at large
scales ($\ell \gg \ell_{0}$). The appropriately defined effective dimension,
$D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg
\ell_{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell_{0}$). This suggests that the
physical spacetime becomes essentially 2-dimensional near Planck scale.
