## Realization Spaces of 4-Polytopes are Universal (1995)

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Venue: | BULL. AMER. MATH. SOC |

Citations: | 13 - 4 self |

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@ARTICLE{Richter-Gebert95realizationspaces,

author = {Jürgen Richter-Gebert and Günter M. Ziegler},

title = {Realization Spaces of 4-Polytopes are Universal},

journal = {BULL. AMER. MATH. SOC},

year = {1995},

volume = {32},

pages = {403--412}

}

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### Abstract

Let P be a d-dimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report

### Citations

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Citation Context ... polytopes. Duality theory (“Gale diagrams” [9, 23]) was used to construct a non-rational 8-polytope with 12 vertices (Perles, 1967 [9]). Also Mnëv’s famous Universality Theorem for oriented matroids =-=[15, 16, 5, 10]-=- via Gale diagrams implies a universality theorem for d-polytopes with d + 4 vertices: in general for such polytopes the realization spaces can be arbitrarily complicated.4 JÜRGEN RICHTER-GEBERT AND ... |

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Citation Context ...eeded to coordinatize the vertex set of a given d-dimensional polytope? How can one tell whether a finite lattice is the face lattice of a polytope or not? For 3-dimensional polytopes, Steinitz’ wor=-=k [19, 20] answere-=-d these basic questions about realization spaces more then seventy years ago. In particular, Steinitz’ “Fundamentalsatz der konvexen Typen” (today known as Steinitz’ Theorem) and its modern re... |

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Citation Context ...l polytopes. Duality theory (“Gale diagrams” [9, 23]) was used to construct a non-rational 8-polytope with 12 vertices (Perles 1967 [9]). Also Mnëv’s famous Universality Theorem for oriented ma=-=troids [15, 16, 5, 11] vi-=-a Gale diagrams implies a universality theorem for d-polytopes with d + 4 vertices: in general for such polytopes the realization spaces can be arbitrarily complicated. Mnëv’s Universality Theorem ... |

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Citation Context ... basic questions about realization spaces more then seventy years ago. In particular, Steinitz’s “Fundamentalsatz der konvexen Typen” (today known as Steinitz’s Theorem) and its modern relatives (see =-=[9]-=- and [23]) provide complete answers to these questions for this special case. Steinitz’s Theorem, 1922. A graph G is the edge graph of a 3-polytope if and only if G is simple, planar and 3-connected. ... |

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Citation Context ...eded to coordinatize the vertex set of a given d-dimensional polytope? How can one tell whether a finite lattice is the face lattice of a polytope or not? For 3-dimensional polytopes, Steinitz’s work =-=[19, 20]-=- answered these basic questions about realization spaces more then seventy years ago. In particular, Steinitz’s “Fundamentalsatz der konvexen Typen” (today known as Steinitz’s Theorem) and its modern ... |

22 |
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Citation Context ...f The proof of the Main Theorem is constructive. It starts with the defining equations of a semialgebraic set, and uses them to explicitly construct the face lattice of a 4-polytope. A result of Shor =-=[18] is -=-used, which states that every primary semialgebraic set V is stably equivalent to a semialgebraic set V ′ ∈ R n whose defining inequalities fix a total order 1 = x1 < x2 < x3 < . . . < xn on the v... |

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Citation Context ...systematic construction methods for d-polytopes in any fixed dimension d. Only a “sporadic” example of a 4-polytope with disconnected realization space was constructed (Bokowski, Ewald & Kleinschm=-=idt [6, 7, 15]-=-). More sporadic examples showed that the shapes of 3-faces of 4-polytopes (Kleinschmidt [12], Barnette [1]) and of 2-faces of 5-polytopes (Ziegler [23]), cannot be prescribed arbitrarily. Until now n... |

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Citation Context ...: it can be “reconstructed” as the intersection of the line spanned by the two new points with the d-hyperplane spanned by the original point configuration. The “classical” use of Lawrence ext=-=ensions [4, 15]-=- starts with a 2-dimensional configuration of n points, and performs Lawrence extensions on all these points, one after the other. The resulting configuration of 2n points is the vertex set of an (n +... |

10 |
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Citation Context ...rete geometry, such spaces appear in subjects as diverse as algebraic geometry (moduli spaces), differential topology (see Cairns’ smoothing theory [8]), and nonlinear optimization (see Günzel et a=-=l. [11]-=-). Assume that in Definition 1 each point p i for i = 1, . . . , n is a vertex of P . A realization of a polytope P is a polytope Q = conv(q 1, . . . , q n) such that the face lattices of P and Q are ... |

10 | The triangulations of the 3-sphere with up to 8 vertices - Barnette - 1973 |

9 |
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Citation Context ...eeded to coordinatize the vertex set of a given d-dimensional polytope? How can one tell whether a finite lattice is the face lattice of a polytope or not? For 3-dimensional polytopes, Steinitz’ wor=-=k [19, 20] answere-=-d these basic questions about realization spaces more then seventy years ago. In particular, Steinitz’ “Fundamentalsatz der konvexen Typen” (today known as Steinitz’ Theorem) and its modern re... |

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Citation Context ...cribed, that is, the canonical map R(P ) → R(F ) is surjective for every facet F ⊆ P (Barnette & Grünbaum [2]). Similar statements for d-polytopes that have at most d + 3 vertices were proved by =-=Mani [14] and-=- Kleinschmidt [13]. Over the years, it became clear that no similar positive answer could be expected for high-dimensional polytopes. Duality theory (“Gale diagrams” [9, 23]) was used to construct... |

7 |
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Citation Context ...ordinates. • The shape of one 2-face in the boundary of a 3-polytope P can be arbitrarily prescribed, that is, the canonical map R(P ) → R(F ) is surjective for every facet F ⊆ P (Barnette & Gr�=-=�nbaum [2]-=-). Similar statements for d-polytopes that have at most d + 3 vertices were proved by Mani [14] and Kleinschmidt [13]. Over the years, it became clear that no similar positive answer could be expected... |

6 |
The universality theorems on the oriented matroid stratification of the space of real matrices
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Citation Context ...l polytopes. Duality theory (“Gale diagrams” [9, 23]) was used to construct a non-rational 8-polytope with 12 vertices (Perles 1967 [9]). Also Mnëv’s famous Universality Theorem for oriented ma=-=troids [15, 16, 5, 11] vi-=-a Gale diagrams implies a universality theorem for d-polytopes with d + 4 vertices: in general for such polytopes the realization spaces can be arbitrarily complicated. Mnëv’s Universality Theorem ... |

6 | The complete enumeration of the 4polytopes and 3-spheres with eight vertices - Altshuler, Steinberg - 1985 |

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On facets with non-arbitrary shapes
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Citation Context ... of a 4-polytope with disconnected realization space was constructed (Bokowski, Ewald & Kleinschmidt [6, 7, 15]). More sporadic examples showed that the shapes of 3-faces of 4-polytopes (Kleinschmidt =-=[12]-=-, Barnette [1]) and of 2-faces of 5-polytopes (Ziegler [23]), cannot be prescribed arbitrarily. Until now no general construction techniques to produce polytopes with controllably bad behavior for any... |

4 | Uniform oriented matroids without the isotopy property, Discrete Comput - Jaggi, Mani-Levitska, et al. - 1989 |

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Two “simple” 3-spheres
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Citation Context ...a general semialgebraic set one also admits non-strict inequalities hk ≥ 0. Thus, for example, the set {0, 1} and the open interval ]0, 1[⊂ R are primary semialgebraic sets, while the closed inter=-=val [0, 1]-=- is a semialgebraic set in R that is not primary. In this research report we present a Universality Theorem proved by the first author [17], stating that all primary semialgebraic sets are in a suitab... |

3 |
de Oliveira, Simplicial convex 4-polytopes do not have the isotopy property
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Citation Context ...systematic construction methods for d-polytopes in any fixed dimension d. Only a “sporadic” example of a 4-polytope with disconnected realization space was constructed (Bokowski, Ewald & Kleinschm=-=idt [6, 7, 15]-=-). More sporadic examples showed that the shapes of 3-faces of 4-polytopes (Kleinschmidt [12], Barnette [1]) and of 2-faces of 5-polytopes (Ziegler [23]), cannot be prescribed arbitrarily. Until now n... |

3 |
Homeomorphisms between topological manifolds and analytic manifolds
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Citation Context ...heir intrinsic importance for questions of real discrete geometry, such spaces appear in subjects as diverse as algebraic geometry (moduli spaces), differential topology (see Cairns’ smoothing theor=-=y [8])-=-, and nonlinear optimization (see Günzel et al. [11]). Assume that in Definition 1 each point p i for i = 1, . . . , n is a vertex of P . A realization of a polytope P is a polytope Q = conv(q 1, . .... |

3 | Boundary complexes of convex polytopes cannot be characterized locally
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Citation Context ...annot be prescribed arbitrarily. Until now no general construction techniques to produce polytopes with controllably bad behavior for any fixed dimension d were known. The σ-construction presented in=-= [21] f-=-or that purpose turned out to be incorrect [23]. In the following we report on the first author’s recent work [17] that produces a complete systematic Universality Theorem for polytopes of dimension... |

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3 |
mit wenigen Ecken, Geom
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Citation Context ... canonical map R(P) → R(F) is surjective for every facet F ⊆ P (Barnette & Grünbaum [2]). Similar statements for d-polytopes that have at most d + 3 vertices were proved by Mani [14] and Kleinschmidt =-=[13]-=-. Over the years, it became increasingly clear that no similar positive answer could be expected for high-dimensional polytopes. Duality theory (“Gale diagrams” [9, 23]) was used to construct a non-ra... |

2 | Neuere Entwicklungen in der kombinatorischen Konvexgeometrie, in: Contributions to Geometry - Ewald, Kleinschmidt, et al. - 1979 |

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2 |
Multiparametric optimization: On stable singularities occurring in combinatorial partition codes, Control Cybernet
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Citation Context ... we use to compare realization spaces with general primary semialgebraic sets is stable equivalence. Although such a concept has been used by different authors, the precise definitions they used (see =-=[11, 10, 15, 16, 18]-=-) vary substantially in their technical content. The common idea is that semialgebraic sets that only differ by a “trivial fibration” and a rational change of coordinates should be considered as stabl... |

2 |
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Citation Context ... at least a doubly exponential function in n .UNIVERSALITY FOR POLYTOPES 5 In particular these implications solve all the problems that were recently emphasized in “Three problems about 4-polytopes” =-=[22]-=-. They also solve [23, Problems 5.11 ∗ , 6.10 ∗ , and 6.11 ∗ ]. 3. Stable equivalence The concept we use to compare realization spaces with general primary semialgebraic sets is stable equivalence. Al... |

2 |
on polytopes, Graduate Texts in Math., vol. 152
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(Show Context)
Citation Context ...uestions about realization spaces more then seventy years ago. In particular, Steinitz’s “Fundamentalsatz der konvexen Typen” (today known as Steinitz’s Theorem) and its modern relatives (see [9] and =-=[23]-=-) provide complete answers to these questions for this special case. Steinitz’s Theorem, 1922. A graph G is the edge graph of a 3-polytope if and only if G is simple, planar and 3-connected. Here simp... |

1 |
Convex Polytopes, Interscience Publ., London 1967; revised edition
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(Show Context)
Citation Context ...se basic questions about realization spaces more then seventy years ago. In particular, Steinitz’ “Fundamentalsatz der konvexen Typen” (today known as Steinitz’ Theorem) and its modern relativ=-=es (see [9] a-=-nd [23]) provide complete answers to these questions for this special case. Steinitz’ Theorem, 1922. A graph G is the edge graph of a 3-polytope if and only if G is simple, planar and 3-connected. H... |

1 |
Hirabayashi & H.Th. Jongen, Multiparametric optimization: On stable singularities occurring in combinatorial partition codes
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(Show Context)
Citation Context ... we use to compare realization spaces with general primary semialgebraic sets is stable equivalence. Although such a concept has been used by different authors, the precise definitions they used (see =-=[10, 11, 15, 16, 18]) va-=-ry substantially in their technical content. The common idea is that semialgebraic sets that only differ by a “trivial fibration” and a rational change of coordinates should be considered as stabl... |