Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

Karim A. Adiprasito, Arnau Padrol*, Louis Theran

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of Q are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

Original languageAmerican English
Pages (from-to)412-431
Number of pages20
JournalDiscrete and Computational Geometry
Issue number2
StatePublished - 27 Sep 2015

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.


  • Delaunay triangulation
  • Inscribed polytope
  • Realization space
  • Universality theorem


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