  The general aim of manifold estimation is reconstructing, by statistical
methods, an $m$-dimensional compact manifold $S$ on ${\mathbb R}^d$ (with
$m\leq d$) or estimating some relevant quantities related to the geometric
properties of $S$. We will assume that the sample data are given by the
distances to the $(d-1)$-dimensional manifold $S$ from points randomly chosen
on a band surrounding $S$, with $d=2$ and $d=3$. The point in this paper is to
show that, if $S$ belongs to a wide class of compact sets (which we call \it
sets with polynomial volume\rm), the proposed statistical model leads to a
relatively simple parametric formulation. In this setup, standard methodologies
(method of moments, maximum likelihood) can be used to estimate some
interesting geometric parameters, including curvatures and Euler
characteristic. We will particularly focus on the estimation of the
$(d-1)$-dimensional boundary measure (in Minkowski's sense) of $S$.
  It turns out, however, that the estimation problem is not straightforward
since the standard estimators show a remarkably pathological behavior: while
they are consistent and asymptotically normal, their expectations are infinite.
The theoretical and practical consequences of this fact are discussed in some
detail.
