  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.
