  For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert
x-x_i\rVert^2/2h^2)$ associated to a point cloud
$\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of
closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a
continuous version of an alpha shape. Using arguments based on Fenchel duality,
we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set
$\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(a)$ can be
realized as the union of a certain power-shifted covering by balls with centers
in $\mathcal{S}(a)$. By extracting finite alpha complexes with vertices in
$\mathcal{S}(a)$, we obtain refined geometric models of noisy point clouds, as
well as density-filtered persistent homology calculations. In order to compute
alpha complexes in higher dimension, we used a recent algorithm due to the
present authors based on the duality principle.
