Variance of the distance to the boundary of convex domains in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$

TL;DR

This paper systematically studies the variance of the distance to the boundary in convex domains in 2D and 3D, introducing the variocentre in 2D and analyzing variance-distance relationships in 3D.

Contribution

It introduces the variocentre in 2D as a new domain center concept and explores variance-bound relationships in 3D, connecting to biological classification algorithms.

Findings

Variance function in 2D is strictly convex.

Introduction of the variocentre as a new domain center.

Mathematical justification for boundary classification algorithms.

Abstract

In this paper, we give for the first time a systematic study of the variance of the distance to the boundary for arbitrary bounded convex domains in $\mathbb{R}^2$ and $\mathbb{R}^3$. In dimension two, we show that this function is strictly convex, which leads to a new notion of the centre of such a domain, called the variocentre. In dimension three, we investigate the relationship between the variance and the distance to the boundary, which mathematically justifies claims made for a recently developed algorithm for classifying interior and exterior points with applications in biology.