2024 On the notion of polynomial reach: A statistical application
Alejandro Cholaquidis, Antonio Cuevas, Leonardo Moreno
Electron. J. Statist. 18(2): 3437-3460 (2024). DOI: 10.1214/24-EJS2278

## Abstract

The volume function $V\left(t\right)$ of a compact set $S\in {\mathbb{R}}^{d}$ is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called “positive reach”) which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V\left(t\right)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski’s sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call “polynomial reach”) might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach, or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points. This paper explores the theoretical and practical aspects of this idea.

## Funding Statement

The research of the first and third authors has been partially supported by grant FCE-3-2022-1-172289 from ANII (Uruguay), 22MATH-07 form MATH – AmSud (France-Uruguay) and 22520220100031UD from CSIC (Uruguay). The research of the second author has been partially supported by Grants PID2019-109387GB-I00 from the Spanish Ministry of Science and Innovation and Grant CEX2019-000904-S funded by MCIN/AEI/ 10.13039/501100011033.

## Acknowledgments

The authors would like to express their gratitude to the Associate Editor and the Editor for their constructive comments, which significantly enhanced the quality of this paper. They also acknowledge with appreciation the valuable feedback from the two anonymous reviewers.

## Citation

Alejandro Cholaquidis. Antonio Cuevas. Leonardo Moreno. "On the notion of polynomial reach: A statistical application." Electron. J. Statist. 18 (2) 3437 - 3460, 2024. https://doi.org/10.1214/24-EJS2278

## Information

Received: 1 July 2023; Published: 2024
First available in Project Euclid: 27 August 2024

arXiv: 2010.00000
Digital Object Identifier: 10.1214/24-EJS2278

Subjects:
Primary: 62G05 , 62G20
Secondary: 60D05 , 68W25

Keywords: condition number , Minkowski content , polynomial volume , reach , volume function

JOURNAL ARTICLE
24 PAGES

Vol.18 • No. 2 • 2024