Delaunay triangulations with disconnected realization spaces

Abstract

We study realization spaces of Delaunay triangulations and show that they can be arbitrarily complicated, and in particular disconnected. Our smallest example consists of two configurations of 29 labeled points in R25 whose Delaunay triangulations are combinatorially equivalent but yet there is no continuous transformation that maps one to the other without changing the triangulation. In general, we prove that the realization space of a Delaunay triangulation in Rd can have Ω(2d) connected components. Our proof uses Mnev's Universality Theorem and also shows that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.