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.

翻译：设 $S$ 为 $\mathbb{R}^d$ 中处于一般位置的 $n$ 个点集。$S$ 的 $k$ 阶Voronoi图 $V_k(S)$ 是对 $\mathbb{R}^d$ 的剖分，使得每个胞腔内的点具有相同的 $k$ 个最近邻点。Sibson在其1980年开创性论文（《Dirichlet镶嵌的向量恒等式》）中，通过利用 $V_2(S)$ 胞腔与 $V_1(S)$ 中 $Q$ 的胞腔交集体积比率，给出了将 $S$ 中某点 $Q$ 表示为 $S$ 中其他点凸组合的公式。自然邻点插值法即基于Sibson公式。我们将这一结果推广至任意阶Voronoi图，通过体积比率将 $Q$ 表示为 $S$ 中其他点的凸组合。