Polynomial volume estimation and its applications

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 given.