Sibson's formula for higher order Voronoi diagrams
<|reference_start|>Sibson's formula for higher order Voronoi diagrams: Let $S$ be a set of $n$ points in general position in $\mathbb{R}^d$. The order-$k$ Voronoi diagram of $S$, $V_k(S)$, is a subdivision of $\mathbb{R}^d$ 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 $S$ by using ratios of volumes of the intersection of cells of $V_2(S)$ and the cell of $Q$ in $V_1(S)$. 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.<|reference_end|>
@article{claverol2024sibson's,
    title={Sibson's formula for higher order Voronoi diagrams},
    author={Merc`e Claverol, Andrea de las Heras-Parrilla, Clemens Huemer and
  Dolores Lara},
    journal={arXiv preprint arXiv:2404.17422},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.17422},
    primaryClass={cs.CG}
}