  An exact computation of the persistent Betti numbers of a submanifold $X$ of
a Euclidean space is possible only in a theoretical setting. In practical
situations, only a finite sample of $X$ is available. We show that, under
suitable density conditions, it is possible to estimate the multidimensional
persistent Betti numbers of $X$ from the ones of a union of balls centered on
the sample points; this even yields the exact value in restricted areas of the
domain.
  Using these inequalities we improve a previous lower bound for the natural
pseudodistance to assess dissimilarity between the shapes of two objects from a
sampling of them.
  Similar inequalities are proved for the multidimensional persistent Betti
numbers of the ball union and the one of a combinatorial description of it.
