# Realization spaces of 4-polytopes are universal

@article{RichterGebert1995RealizationSO, title={Realization spaces of 4-polytopes are universal}, author={J{\"u}rgen Richter-Gebert and G{\"u}nter M. Ziegler}, journal={Bulletin of the American Mathematical Society}, year={1995}, volume={32}, pages={403-412} }

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V…

## 19 Citations

### A Universality Theorem for Nested Polytopes

- MathematicsArXiv
- 2019

It is shown that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent, and that unless $\exists \mathbb{R} =$ NP, the NPP is not contained in the complexity class NP.

### $\forall \exists \mathbb{R}$-completeness and area-universality

- Mathematics
- 2017

This work conjecture that the problem Area Universality is $\forall\exists \mathbb{R}$-complete and support this conjecture by a series of partial results, where it is proved that $\exists Â£R$- and $\ forall\Exists $-completeness of variants of Area Universalities$ are true.

### A universality theorem for allowable sequences with applications

- MathematicsArXiv
- 2018

This work shows that the realization spaces of allowable sequences are universal and consequently deciding the realizability is complete in the existential theory of the reals (\ER), and argues that this result is a useful tool for further geometric reductions.

### A Quantitative Steinitz Theorem for Plane Triangulations

- MathematicsArXiv
- 2013

It is proved that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface.

### A universality theorem for nonnegative matrix factorizations

- Mathematics, Computer Science
- 2016

It is shown that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization.

### The Art Gallery Problem is $\exists \mathbb{R}$-complete

- Mathematics
- 2017

It is proved that the art gallery problem is $\exists \mathbb{R}$-complete, implying that any system of polynomial equations over the real numbers can be encoded as an instance of the art Gallery problem.

### Covering Polygons is Even Harder

- Mathematics, Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

It is proved that assuming the widespread belief that NP-hard MCC is not in N P, and the problem is thus $\exists \mathbb{R}$-complete, that many natural approaches to finding small covers are bound to give suboptimal solutions in some cases.

### A Practical Algorithm with Performance Guarantees for the Art~Gallery Problem

- Computer ScienceSoCG
- 2021

A one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program guarantees to find the optimal solution for every vision stable polygon.

### Embeddings of Polytopes and Polyhedral Complexes

- Mathematics
- 2012

Author(s): Wilson, Stedman | Advisor(s): Pak, Igor | Abstract: When does a topological polyhedral complex (embedded in Rd) admit a geometric realization (a rectilinear embedding in Rd)? What are the…

### Smoothed Analysis of Order Types

- MathematicsArXiv
- 2019

The results show that order type realizability is much easier for realistic instances than in the worst case, and one of the first $\exists\mathbb{R}$-complete problems analyzed under the lens of Smoothed Analysis can recognize instances in "expected \NP-time".

## References

SHOWING 1-10 OF 68 REFERENCES

### THREE PROBLEMS ABOUT 4-POLYTOPES

- Mathematics
- 1994

To some extent, we can claim to“understand” 3-dimensional polytopes. in fact, Steinitz’ Theorem
“the combinatorial types of 3-polytopes are given by the simple, 3-connected planar graphs”…

### The number of polytopes, configurations and real matroids

- Mathematics
- 1986

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/ oo. A similar bound for the number of simplicial polytopes has previously been proved by Goodman…

### Some Applications of Affine Gale Diagrams to Polytopes with few Vertices

- MathematicsSIAM J. Discret. Math.
- 1988

A new negative Steinitz-type theorem is established; the face lattices of simplicial k-polytopes with $k + 4$ vertices cannot be characterized locally, and the affine Gale diagrams corresponding to a given simplicial complex are characterized.

### Polytopal and nonpolytopal spheres an algorithmic approach

- Mathematics
- 1987

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which,…

### The universal partition theorem for oriented matroids

- MathematicsDiscret. Comput. Geom.
- 1996

This work presents the first proof of the Universal Partition Theorem, and includes the first completeProof of the so-called Universality Theorem.

### Boundary Complexes of Convex Polytopes cannot Be Characterized Locally

- Mathematics
- 1987

It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the set-up of chirotopes or oriented matroids to derive a similar result in…

### Computational Synthetic Geometry

- Mathematics
- 1989

Computational synthetic geometry aims to develop algorithms to find for a given abstract geometric object either a coordinatization over some field or a proof that such a realization does not exist.…

### Lectures on Polytopes

- Mathematics
- 1994

Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward…

### Lawrence Polytopes

- MathematicsCanadian Journal of Mathematics
- 1990

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is…

### Shadow-boundaries and cuts of convex polytopes

- Mathematics
- 1980

Let P be a (convex) d -polytope in the Euclidean space E d and p a point of E d not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of…