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Universality theorems for inscribed polytopes and Delaunay triangulations by Adiprasito, Karim A.; Padrol, Arnau; Theran, Louis is a scholarly article available to read on EtoBox.

What is Universality theorems for inscribed polytopes and Delaunay triangulations about?

We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to M\"obius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mn\"ev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

Author

Adiprasito, Karim A.; Padrol, Arnau; Theran, Louis

Published

2014

Language

EN