Sibson's formula for higher order Voronoi diagrams

Let S be a set of n points in general position in $R^d$. The order-k Voronoi diagram of S, $V_k\left(S\right)$, is a subdivision of $R^d$ into cells whose points have the same k nearest points of S. Sibson, in his seminal paper from 1980 (A vector identity for the Dirichlet tessellation), gives a formula to express a point Q of S as a convex combination of other points of S using ratios of volumes of the intersection of cells of $V_2\left(S\right)$ and the cell of Q in $V_1\left(S\right)$. The natural neighbour interpolation method is based on Sibson's formula. We generalize his result to express Q as a convex combination of other points of S by using ratios of volumes from Voronoi diagrams of any given order.