  We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic
Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$,
where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the
half-space given by $\langle \xi, \nu\rangle > d$ this generalizes an
inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the
inequality is sharp and takes the form \begin{equation}
  \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega}
\sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi),
\nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi,
\end{equation} where $\textrm{dist}(\, \cdot\,, \partial \Omega)$ denotes the
Euclidean distance from $\partial \Omega$.
