  The Valeriepieris circle is the smallest circle that can be draw on the globe
containing half of the world's population. The Valeriepieris (VP) circle acts
as a spatial median, effectively splitting spatial data into two halves in a
unique way. In this paper the idea of the VP circle is generalised and a fast
algorithm and corresponding software package to compute it are described. The
VP circle is compared to other measures of centre and dispersion for population
distributions and is shown to reflect expected differences between countries
and changes over time. By studying the VP circle as a function of the included
population fraction a new way of representing population distributions is
constructed, as well as a mathematical model of its expected behaviour. Finally
a measure of population `centralisation' is constructed which measures the
tendency of a territory to be dominated by a single population centre or to
have a more even distribution of population.
