I am reading "Laguerre Voronoi Diagram on the Sphere" by Kokichi Sugihara (I have provided a link to the paper at the bottom of this post). In this paper the author defines a notion of Laguerre Voronoi diagram on the sphere (also called a power diagram). One can define a power diagram in $\mathbb{R}^d$ by starting with a set of $n$ points $p_i$ equipped with weights $w_i$, and decomposing $\mathbb{R}^d$ into cells $$W_i=\{x\in \mathbb{R}^d | d(x,p_i)-w_i < d(x,p_j)-w_j, \forall j\neq i\}$$ (for those familiar with power diagrams there is another way to phrase the weights in terms of radii of circles centered at points $p_i$, where $w_i=r_i^2$). The intuition is that (like Voronoi diagrams) the cell $W_i$ corresponds to all $x\in \mathbb{R}^d$ such that $x$ is closer to $p_i$ than any other $p_j$ in our set of points, where closeness is defined with respect to $d(x,p_i)-w_i$. In particular notice that this is a generalization of Voronoi diagrams, where if all $w_i=0$ we recover the Voronoi decomposition of $\mathbb{R}^d$.
Sugihara defines a version of this construction on the unit 2-sphere, where now proximity between a point $p_i$ in our defining set $\{p_i\}_{i=1}^n$ and an arbitrary point $p\in S^2$ is given by $d_L(p_i,p)$, which is defined to be proportional to $\cos(d(p_i,p))$ where $d$ is the geodesic distance on the surface of the sphere. I am suppressing the dependence of $d_L$ on a set of weights, but for the purposes of my question, we can assume all weights are equal (say to 0), in which case one would think we would recover the Voronoi diagram. Once again, these diagrams are defined by sets $$W_i=\{p\in S^2|d_L(p_i,p)<d_L(p_j,p)\}.$$
Here is my question: Since $d_L(p_i,p)$ is proportional with $\cos(d(p_i,p))$, it is inversely proportional with $d(p_i,p)$, since the maximum distance between two points on the unit sphere is $\pi$, and cosine is monotonically decreasing in $[0,\pi]$. But then since $d_L$ decreases as distance increases, and the sets $W_i$ are defined by minimizing $d_L$, $W_i$ should consist of all $p\in S^2$ that are further away from $p_i$ than all other $p_j$ in terms of the geodesic distance on the sphere. This seems to be a very unintuitive definition, and at odds with Euclidean Voronoi/Power diagrams, where $W_i$ consists of points which are close to $p_i$.
Am I missing something? Why would one want to define a version of Voronoi diagrams by points that are further away, rather than closer? Or have I made an error in reasoning somewhere?
Link to the paper in question: https://www.heldermann-verlag.de/jgg/jgg06/jgg0606.pdf
It appears there is an error in the sign of the inequality in equation (14). Sugihara notes that the "Laguerre Proximity" (this expression involving the cosine) is inversely related to distance; it decreases as we move away from point / circle.
In each of the results(e.g., the definition of the Laguerre bisector in (8), Theorem 1), the Laguerre proximity is appears in equality expressions. Sugihara is analyzing the boundaries between Laguerre Voronoi cells. Everything is ok except the sign of the inequality in the definition of the Laguerre Voronoi definition is backwards: to be "nearer" a circle means to that a larger (not smaller) Laguerre proximity.