  Given a smooth submanifold of the Euclidean space, a finite point cloud and a
scale parameter, we introduce a construction which we call the flat Delaunay
complex (FDC). This is a variant of the tangential Delaunay complex (TDC)
introduced by Boissonnat et al.. Building on their work, we provide a short and
direct proof that when the point cloud samples sufficiently nicely the
submanifold and is sufficiently safe (a notion which we define in the paper),
our construction is homeomorphic to the submanifold. Because the proof works
even when data points are noisy, this allows us to propose a perturbation
scheme that takes as input a point cloud sufficiently nice and returns a point
cloud which in addition is sufficiently safe. Equally importantly, our
construction provides the framework underlying a variational formulation of the
reconstruction problem which we present in a companion paper.
