The universality theorem for neighborly polytopes

In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an even-dimensional neighborly polytope. This in particular provides the final step for Mn\"ev's proof of the universality theorem for simplicial polytopes.Comment: 5 pages, 1 figure. Small change

Universality theorems for inscribed polytopes and Delaunay triangulations

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.Comment: 15 pages, 2 figure

Six topics on inscribable polytopes

Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six particular topics related to inscribable polytopes. Along the way we collect a list of (new and old) open questions.Comment: 11 page

Positive Plücker tree certificates for non-realizability

We introduce a new method for finding a non-realizability certificate of a simplicial sphere S: we exhibit a monomial combination of classical 3-term Pl¨ucker relations that yields a sum of products of determinants that are known to be positive in any realization of S; but their sum should vanish, contradiction. Using this technique, we prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere constructed by Zheng, a family of highly neighborly centrally symmetric spheres constructed by by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method in fact works for orientable pseudo-manifolds, not just for spheres.Peer ReviewedPostprint (author's final draft