Let $p : n \rightarrow \mathbb{R}^d$ be a sequence of points. Let $\lambda : \mathbb{R}^d \rightarrow \mathbb{R}^n$ be a generalized barycentric coordinate system (GBCS) with respect to $p$. That is, \begin{align*} 1 &= \sum_{i \in n} \lambda(x)_i \\ x &= \sum_{i \in n} \lambda(x)_i p_i \end{align*} for all $x \in \mathbb{R}^d$. Affine generalised barycentric coordinates by Shayne Waldron defines a GBCS with nice properties by imposing the additional constraint of minimizing the $L_2$ norm \begin{align*} \|\lambda(x)\|_2^2 = \sum_{i \in n} |\lambda(x)_i|^2 \end{align*} at each $x \in \mathbb{R}^d$. However, this GBCS doesn't satisfy the Lagrange property, namely that \begin{align*} \lambda(p_j)_i = \delta_{ji} \end{align*} for all $i, j \in n$. Is there a "nice" continuous GBCS which satisfies this property?
I think adding the constraint of minimizing the $L_0$ "norm" (i.e. maximizing sparsity) guarantees the Lagrange property, but does it guarantee uniqueness everywhere else?
Generalized barycentric coordinates in computer graphics and computational mechanics by Kai Hormann and N. Sukumar lists (on Table 1.1, page 15) 3 methods that are defined with respect to sets of scattered points, as opposed to just convex polygons or arbitrary simple polygons:
Affine coordinates (Waldron).
Sibson natural neighbor coordinates.
Laplace coordinates.
Of these, only the last two satisfy the Lagrange property, and they require the computation of Voronoi diagrams. Another approach I can think of is to construct the Delaunay triangulation of the point set and then use ordinary barycentric coordinates on the triangle a point falls inside of.