  Consider a planar random point process made of the union of a point (the
origin) and of a Poisson point process with a uniform intensity outside a
deterministic set surrounding the origin. When the intensity goes to infinity,
we show that the Voronoi cell associated with the origin converges from above
to a deterministic convex set. We describe this set and give the asymptotics of
the expectation of its defect area, defect perimeter and number of vertices. On
the way, two intermediary questions are treated. First, we describe the mean
characteristics of the Poisson-Voronoi cell conditioned on containing a fixed
convex body around the origin and secondly, we show that the nucleus of such
cell converges to the Steiner point of the convex body. As in R\'enyi and
Sulanke's seminal papers on random convex hulls, the regularity of the convex
body has crucial importance. We deal with both the smooth and polygonal cases.
Techniques are based notably on accurate estimates of the area of the Voronoi
flower and of the support function of the cell containing the origin as well as
on an Efron-type relation.
