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We prove geometric $L^p$ versions of Hardy's inequality for the sub-ellipticLaplacian 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 thehalf-space given by $\langle \xi, \nu\rangle > d$ this generalizes aninequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ theinequality 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 theEuclidean distance from $\partial \Omega$.

Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group