  In this paper, we investigate the similarity property of the sample
projection depth contours. It turns out that some of these contours are of
\emph{the same shape} with different sizes, following a similar fashion to the
Mahalanobis depth contours. One advantage of this investigation is the
potential of bringing convenience to the computation of the projection depth
contours; the other one is that we may utilize this idea to extend both the
halfspace depth and zonoid depth to versions that do not vanish outside the
convex hull of the data cloud, aiming at overcoming the so-called `outside
problem'. Examples are also provided to illustrate the main results.
