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

Alastair N. Fletcher; Alexander G. Fletcher

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arXiv · 2407.12041

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

Alastair N. Fletcher·Alexander G. Fletcher

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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.

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BibTeX RIS

Alastair N. Fletcher, Alexander G. Fletcher. 2024-07-01. Variance of the distance to the boundary of convex domains in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. https://arxiv.org/abs/2407.12041

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