Given a compact set S ⊂ R
d we consider the problem of estimating, from a random sample of points, the Lebesgue measure of S, µ(S), and its boundary measure, L(S) (as defined
by the Minkowski content of ∂S). This topic has received some attention, especially in the
two-dimensional case d = 2, motivated by applications in image analysis. A new method to
simultaneously estimate µ(S) and L(S) from a sample of points inside S is proposed.
The basic idea is to assume that S has a polynomial volume, that is, that V (r) := µ{x :
d(x, S) ≤ r} is a polynomial in r of degree d, for all r in some interval [0, R). We develop a
minimum distance approach to estimate the coefficients of V (r) and, in particular µ(S) and L(S),
which correspond, respectively, to the independent term and the first degree coefficient of V (r).
The strong consistency of the proposed estimators is proved. Some numerical illustrations are
givenThis work has been partially supported by Spanish Grants MTM2016-78751-P (A. Cuevas) and MTM2016-76969-P (B. Pateiro-López)S
