On the Embeddability of Delaunay Triangulations in Anisotropic, Normed, and Bregman Spaces

TL;DR

This paper proves that in certain convex divergence spaces, orphan-free Voronoi diagrams have connected edges and vertices, and their duals form embedded triangulations, extending known results to anisotropic, Bregman, and normed spaces.

Contribution

It establishes conditions under which Voronoi diagrams and their duals are embedded in anisotropic, Bregman, and normed divergence spaces, generalizing classical Euclidean results.

Findings

Voronoi edges and vertices are connected under orphan-freedom.

Dual graphs of orphan-free diagrams are embedded triangulations.

Results apply to Bregman, anisotropic, and all strictly convex norm-derived divergences.

Abstract

Given a two-dimensional space endowed with a divergence function that is convex in the first argument, continuously differentiable in the second, and satisfies suitable regularity conditions at Voronoi vertices, we show that orphan-freedom (the absence of disconnected Voronoi regions) is sufficient to ensure that Voronoi edges and vertices are also connected, and that the dual is a simple planar graph. We then prove that the straight-edge dual of an orphan-free Voronoi diagram (with sites as the first argument of the divergence) is always an embedded triangulation. Among the divergences covered by our proofs are Bregman divergences, anisotropic divergences, as well as all distances derived from strictly convex $\mathcal{C}^1$ norms (including the $L_p$ norms with $1< p < \infty$). While Bregman diagrams of the {first kind} are simply affine diagrams, and their duals ({weighted} Delaunay triangulations) are always embedded, we show that duals of orphan-free Bregman diagrams of the \emph{second kind} are always embedded.