  We study questions in incidence geometry where the precise position of points
is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced
by narrow tubes, and more generally affine subspaces are replaced by their
small neighborhood. We show that the presence of a sufficiently large number of
approximately collinear triples in a set of points in d dimensional complex
space implies that the points are close to a low dimensional affine subspace.
This can be viewed as a stable variant of the Sylvester-Gallai theorem and its
extensions.
  Building on the recently found connection between Sylvester-Gallai type
theorems and complex Locally Correctable Codes (LCCs), we define the new notion
of stable LCCs, in which the (local) correction procedure can also handle small
perturbations in the euclidean metric. We prove that such stable codes with
constant query complexity do not exist. No impossibility results were known in
any such local setting for more than 2 queries.
