A. Cuevas González
The volume function V(r) of a compact set S in Rd is defined, for r>0, as the Lebesgue measure of the r-parallel set of points within a distance to S not larger than r. From Federer (1959, TAMS) it is known that if S has a positive reach R (the reach of S is the supremum of those values r>0 such that any point within a distance to S smaller than r has a unique metric projection on S) then, V(r) is, for r in [0,R], a polynomial of degree d whose coefficients have interesting geometric interpretations. The minimum distance estimation of V(r) has been considered by Cuevas and Pateiro-López (2018, JSPI) assuming that the parameter R (or at least a lower bound for it) is known and positive.
The estimation of V(r) is addressed here in the light of some recent results on the estimation of the reach parameter (Aamari et al., 2017, EJS; Cholaquidis et al. 2021, arXiv:2110.12208). Some proposals are outlined to combine both problems and some further lines of research are explored.
Keywords: Set estimation, polynomial volume, surface area estimation
Scheduled
RSME-SEIO Invited Session: Topological Data Analysis
June 10, 2022 10:10 AM
A16
