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

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.