### Citations

553 |
Hyperbolic groups, Essays in Group Theory
- Gromov
- 1987
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Citation Context ...nstance by Grünbaum [Grü03]). The notion we use here is more reminiscent of tangent spaces in differential geometry, and in particular more prevalent in geometric group theory (compare for instance =-=[Gro87]-=-). 3 Cross-bedding cubical tori In this section, we define our basic building blocks for the proofs of Theorem I and, ultimately, Theorem II. These building blocks, called cross-bedding cubical tori, ... |

309 |
Introduction to Piecewise-Linear Topology, Springer Study Edition.
- Rourke, Sanderson
- 1982
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Citation Context ... I2 0 00 R(pi/3) 0 0 0 1 . A (geometric) polytopal complex in Rd (resp. in Sd) is a collection of polytopes in Rd (resp. Sd) such that the intersection of any two polytopes is a face of both (cf. =-=[RS72]-=-). Our polytopal complexes are usually finite, i.e. the number of polytopes in the collection is finite. An abstract polytopal complex is a collection of polytopes that are attached along isometries o... |

287 |
The geometry and topology of 3-manifolds,
- Thurston
- 1978
(Show Context)
Citation Context ...ple of a non-inscribable polytope [Ste28], cf. [Grü03, Sec. 13.5]. Much later, interest in inscribed 3-polytopes experienced a revival due to their importance in the theory of hyperbolic 3-manifolds =-=[Thu78]-=- and Delaunay triangulations [Bro79]. Conversely, the connection to hyperbolic geometry led to an almost complete understanding of inscribable polytopes of dimension 3 [Riv96, Riv03]. Many problems co... |

91 | Vorlesungen über die Theorie der Polyeder - Steinitz, Rademacher - 1934 |

87 | The universality theorems on the classification problem of configuration varieties and convex polytope varieties. In - Mnev - 1988 |

79 |
Voronoi diagrams from convex hulls
- Brown
- 1979
(Show Context)
Citation Context ...te28], cf. [Grü03, Sec. 13.5]. Much later, interest in inscribed 3-polytopes experienced a revival due to their importance in the theory of hyperbolic 3-manifolds [Thu78] and Delaunay triangulations =-=[Bro79]-=-. Conversely, the connection to hyperbolic geometry led to an almost complete understanding of inscribable polytopes of dimension 3 [Riv96, Riv03]. Many problems concerning inscribed polytopes remain;... |

52 | A characterization of ideal polyhedra in hyperbolic 3-space’, - Rivin - 1996 |

50 | Realization Spaces of Polytopes
- RICHTER-GEBERT
- 1996
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Citation Context ...ial progress in the understanding of realization spaces of polytopes up to “stable equivalence” (thus, in particular, up to homotopy equivalence), due to the work of Mnëv [Mnë88] and Richter-Gebert =-=[RG96]-=-. Nevertheless, no substantial progress was made on the problem of Perles and Shephard since it was asked in the sixties (see [PS74]). Related results on projectively unique polytopes include: ◦ Any d... |

40 |
Uber krummung und windung geschlossener raumkurven,”
- Fenchel
- 1929
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Citation Context ...C has total curvature less than 2pi. Since the turning number of a closed planar curve is a positive integer multiple of 2pi, the total curvature of C unionsqfat(B,C) C ′ is 2pi. By Fenchel’s Theorem =-=[Fen29]-=-, C unionsqfat(B,C) C ′ is the boundary of a planar convex body. Since every facet is exposed, the boundary complex of this convex body must coincide with C unionsqfat(B,C) C ′ = C ∪ C ′. We proceed t... |

38 | On reciprocal figures and diagrams of forces - Maxwell |

21 | Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionzahlen - Tietze - 1928 |

20 |
Plane self stresses and projected polyhedra i: The basic pattern
- Crapo, Whiteley
- 1993
(Show Context)
Citation Context ...ions and results concerning reciprocal complexes and their convex liftings loosely follows Rybnikov [Ryb00]. For details and an intuitive explanations of the following results, we refer the reader to =-=[CW93]-=- [Aur87] [Ryb00]; more detailed references are collected at the end of this section. All polytopal manifolds in this section are manifolds with boundary. Definition A.2.1 (Duality). Let C be a polytop... |

18 | Cauchy Problem for Integrable Discrete Equations on Quad-Graphs Acta Applicandae Mathematicae 84: 237–262
- Adler, Veselov
- 2004
(Show Context)
Citation Context ...uccessively realizing all the 3-cubes of the abstract cubical 3-complex given by C, starting from those at level 0, amounts to a solution of a Cauchy problem for Q-nets, as studied by Adler & Veselov =-=[AV04]-=- in a “discrete differential geometry” setting. Here the number of initial values, namely the vertex coordinates for the cubes in the layer 0 of C, is still infinite. However, if we divide the standar... |

18 | Neighborly cubical polytopes
- Joswig, Ziegler
(Show Context)
Citation Context ...s for which the dimension of the realization space grows sublinearly with the size are known: In [Zie11], we argued that for the “neighborly cubical polytopes” NCP4[n] constructed by Joswig & Ziegler =-=[JZ00]-=- the dimensions of the realization spaces are low relative to their size: dimRS(NCP4[n]) ∼ (log size NCP4[n])2. Main Results. Trivially, the dimension of RS(P ) for a d-polytope P is always at least d... |

17 |
A Criterion for the Affine Equivalence of Cell Complexes
- Aurenhammer
- 1987
(Show Context)
Citation Context ...d results concerning reciprocal complexes and their convex liftings loosely follows Rybnikov [Ryb00]. For details and an intuitive explanations of the following results, we refer the reader to [CW93] =-=[Aur87]-=- [Ryb00]; more detailed references are collected at the end of this section. All polytopal manifolds in this section are manifolds with boundary. Definition A.2.1 (Duality). Let C be a polytopal d-man... |

17 |
On locally locally convex hypersurfaces with boundary
- Trudinger, Wang
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Citation Context ...s. Van Heijenoort also proved Theorem A.1.1 only for surfaces, but his proof extends to higher dimensions. Expositions of the general case of this theorem and further generalizations are available in =-=[TW02]-=- and [Ryb09]. In this section, we adapt Theorem A.1.1 to polytopal manifolds with boundary. We start off by introducing the notion of immersed polytopal complexes. Definition A.1.2 (Precomplexes). Let... |

13 | Polyeder und Raumeinteilungen, Encyklopädie der mathematischen Wissenschaften - Steinitz - 1922 |

10 |
Graphical statics: two treatises on the graphical calculus and reciprocal figures in graphical statics [Thomas Hudson Beare Trans
- Cremona
(Show Context)
Citation Context ...the existence of the extension. To show this we make use of a relation between reciprocals (or orthogonal duals) and convex liftings of polytopal complexes based on the Maxwell–Cremona correspondence =-=[Cre90]-=- [Max70]. The arguments in this section are only sketched, and there are some substantial disadvantages compared to the approach detailed in the main part of this paper (based on the Alexandrov–van He... |

10 |
Heijenoort, On locally convex manifolds
- van
- 1952
(Show Context)
Citation Context ...vex polytope. A natural corollary of the construction is that the Ts[n] are in locally convex position (i.e. the star of each vertex is in convex position). A theorem of Alexandrov and van Heijenoort =-=[vH52]-=- states that, for d ≥ 3, a locally convex (d− 1)-manifold without boundary in Rd is in fact the boundary of a convex body. As the complexes Ts[n] are manifolds with boundary, we need a version of the ... |

10 |
Systematische Entwicklung der Abhangigkeit geometrischer Gestalten von einander
- Steiner
(Show Context)
Citation Context ...torial types of polytopes that can be realized in an inscribed way are inscribable. Inscribable polytopes are a classical and intriguing subject in polytope theory. Perhaps overly optimistic, Steiner =-=[Ste81]-=- asked in 1832 for a classification of inscribable polytopes. For a long time, it was not even known whether all combinatorial types of polytopes are inscribable, until Steinitz provided an example of... |

9 | Convex polytopes, second ed., Graduate Texts - Grünbaum - 2003 |

9 | Non-rational configurations, polytopes, and surfaces - Ziegler |

8 | Notes on Non-positively Curved Polyhedra, Low dimensional topology - Davis, Moussong - 1996 |

8 | Triangulations with very few geometric bistellar neighbors
- Santos
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Citation Context ...n, that is, in the boundaries of 4-dimensional polytopes. For this our construction will be inspired by the fibration of S3 into Clifford tori, as used in Santos’ work [San00, San12]. We note that in =-=[San00]-=-, the idea to construct polytopal complexes along tori is used to obtain a result related to ours in spirit: Santos provides simplicial complexes that admit only few geometric bistellar flips, whereas... |

7 |
Curves of finite total curvature. In Discrete differential geometry, volume 38 of Oberwolfach Semin
- Sullivan
- 2008
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Citation Context ...nduction on the dimension. First, consider the case d = 2, whose treatment differs from the case d > 2 since Theorem A.1.1 is not applicable. We use the language of curvature of polygonal curves, cf. =-=[Sul08]-=-. If C, C ′ are in S2, use a central projection to transfer C and C ′ to complexes in convex position in R2. If there are two curves C and C ′ in convex position in R2 such that C ′ ∪ fat(B,C) is in c... |

6 |
Polytope skeletons and paths. In Handbook of discrete and computational geometry
- Kalai
- 1997
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Citation Context ...wo questions concerning the spaces of geometric realizations of higher-dimensional polytopes. The first problem originates with Perles and Shephard [PS74]: Problem P-S (Perles & Shephard [PS74] Kalai =-=[Kal04]-=-). Is it true that, for each fixed d ≥ 2, the number of distinct combinatorial types of projectively unique d-polytopes is finite? ∗This work was supported by the DFG within the research training grou... |

6 |
Representations of polytopes and polyhedral sets, Geometriae Dedicata 2
- McMullen
- 1973
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Citation Context ...llows from work of Crapo & Whiteley [CW93, CW94]; compare in particular Theorem 2.6.3 in Rybnikov’s PhD thesis [Ryb00]. A few authors also treated the spherical case directly: in particular, McMullen =-=[McM73]-=- provided a proof for the special case C ∼= Sdeq of Theorem A.2.8 in the spherical setting. The translation of [Ryb00, Thm. 2.6.3] to the spherical case, and hence the proof of Theorems A.2.8 and A.2.... |

5 | Théorèmes de géométrie., Journal de Mathématiques Pures et Appliquées 1re série 3 - Miquel |

3 |
Intrinsic Geometry of Convex Surfaces, OGIZ, Moscow-Leningrad
- Alexandrov
- 1948
(Show Context)
Citation Context ...r at least one x ∈M , there exists a hyperplane H intersecting the convex body Kx only in x). Then M is embedded, and it is the boundary of a convex body. This remarkable theorem is due to Alexandrov =-=[Ale48]-=- in the case of surfaces. Alexandrov did not state it explicitly (his motivation was to prove far stronger results on intrinsic metrics of surfaces), and his proof does not extend to higher dimensions... |

3 |
Polyhedral partitions and stresses
- Rybnikov
(Show Context)
Citation Context ...tori. 29 A.2.1 Duals, reciprocals and liftings to convex position The following summary of basic notions and results concerning reciprocal complexes and their convex liftings loosely follows Rybnikov =-=[Ryb00]-=-. For details and an intuitive explanations of the following results, we refer the reader to [CW93] [Aur87] [Ryb00]; more detailed references are collected at the end of this section. All polytopal ma... |

3 |
Subpolytopes of stack polytopes
- Shephard
- 1974
(Show Context)
Citation Context ...], they are used to prove a universality theorem for projectively unique polytopes and to provide polytopes that are not subpolytopes of any stacked polytope, thus disproving a conjecture of Shephard =-=[She74]-=- and Kalai [Kal04, p. 468] [Kal12]. Finally, in the Appendix we provide the following: ◦ In Section A.1, we discuss the notion of a polytopal complex in (locally) convex position, and establish an ext... |

2 |
Adiprasito, Methods from Differential Geometry in Polytope Theory
- A
- 2013
(Show Context)
Citation Context ...1). Furthermore, we give a tool to check transversality (Proposition 3.3.3). The verification of Propositions 3.3.1 and 3.3.3 is straightforward; their proof can be found in the first author’s thesis =-=[Adi13]-=-. We close the section with a Proposition 3.3.4, which says that the slope is monotone under extensions. Proposition 3.3.1. Let T be a symmetric 2-CCT in S3eq, let v be a vertex of layer 0 of T, and l... |

2 | of stresses, projections, and parallel drawings for spherical polyhedra, Beitraege zur Algebra und Geometrie / Contributions to Algebra and Geometry 35 - Spaces - 1994 |

2 | Inscribable polytopes via Delaunay triangulations
- Gonska
- 2013
(Show Context)
Citation Context ...erbolic geometry led to an almost complete understanding of inscribable polytopes of dimension 3 [Riv96, Riv03]. Many problems concerning inscribed polytopes remain; for some recent progress, compare =-=[Gon13]-=-, [AP14]. In this section, we present some progress in the direction of understanding high-dimensional inscribable polytopes by proving the following analogues of Theorems I and II for inscribed polyt... |

2 |
problems for convex polytopes I’d love to see solved
- Open
(Show Context)
Citation Context ...sality theorem for projectively unique polytopes and to provide polytopes that are not subpolytopes of any stacked polytope, thus disproving a conjecture of Shephard [She74] and Kalai [Kal04, p. 468] =-=[Kal12]-=-. Finally, in the Appendix we provide the following: ◦ In Section A.1, we discuss the notion of a polytopal complex in (locally) convex position, and establish an extension of the Alexandrov–van Heije... |

2 |
von Staudt, Beiträge zur Geometrie der
- C
(Show Context)
Citation Context ...ite family of rational projectively unique d-dimensional polytopes PCCTPQd [n], n ≥ 1, on 12(n+ 1) + d+D − 9 vertices. For the proof, let us first recall a fundamental result going back to von Staudt =-=[vS57]-=-. Proposition A.5.3 (cf. [AP13, Cor. U.17]). Let Q denote any point configuration in Sd+ ⊂ Sd whose elements are described by algebraic coordinates. Then there is a projectively unique point configura... |

1 | The universality theorem for inscribed polytopes and delaunay triangulations, in preparation
- Adiprasito, Padrol
(Show Context)
Citation Context ...eometry led to an almost complete understanding of inscribable polytopes of dimension 3 [Riv96, Riv03]. Many problems concerning inscribed polytopes remain; for some recent progress, compare [Gon13], =-=[AP14]-=-. In this section, we present some progress in the direction of understanding high-dimensional inscribable polytopes by proving the following analogues of Theorems I and II for inscribed polytopes. 36... |

1 |
Éléments de Géométrie, Imprimerie Firmin Didot, Pére et
- Legendre
(Show Context)
Citation Context ...s. 1 Introduction Legendre initiated the study of the spaces of geometric realizations of polytopes, motivated by problems in mechanics. One of the questions studied in his 1794 monograph on geometry =-=[Leg94]-=- is: How many variables are needed to determine a geometric realization of a given (combinatorial type of) polytope? In other words, Legendre asks for the dimension of the realization space RS(P ) of ... |

1 |
reciprocal figures, frames, and diagrams of forces
- On
(Show Context)
Citation Context ...tence of the extension. To show this we make use of a relation between reciprocals (or orthogonal duals) and convex liftings of polytopal complexes based on the Maxwell–Cremona correspondence [Cre90] =-=[Max70]-=-. The arguments in this section are only sketched, and there are some substantial disadvantages compared to the approach detailed in the main part of this paper (based on the Alexandrov–van Heijenoort... |

1 |
Einige Beiträge über konvexe Kurven und Flächen
- Nakajima
- 1931
(Show Context)
Citation Context ...must be strictly smaller than pi, since the contrary assumption would imply that T◦ is the boundary of a convex body in S3eq by Theorem A.1.5, or the more elementary Theorem of Tietze [Tie28, Satz 1] =-=[Nak31]-=-, and in particular homeomorphic to a 2-sphere, contradicting the assumption that T◦ is a torus. Consider the star St(v,T◦) (Figure A.2) for any vertex v of R(T, k − 2). As observed, one of the dihedr... |

1 |
A construction for projectively unique polytopes, Geometriae Dedicata 3
- Perles, Shephard
- 1974
(Show Context)
Citation Context ... + f2(P ) + 4 = f1(P ) + 6. In this paper we treat two questions concerning the spaces of geometric realizations of higher-dimensional polytopes. The first problem originates with Perles and Shephard =-=[PS74]-=-: Problem P-S (Perles & Shephard [PS74] Kalai [Kal04]). Is it true that, for each fixed d ≥ 2, the number of distinct combinatorial types of projectively unique d-polytopes is finite? ∗This work was s... |

1 |
isoperimetrische Probleme bei konvexen Polyedern
- Über
- 1928
(Show Context)
Citation Context ...ication of inscribable polytopes. For a long time, it was not even known whether all combinatorial types of polytopes are inscribable, until Steinitz provided an example of a non-inscribable polytope =-=[Ste28]-=-, cf. [Grü03, Sec. 13.5]. Much later, interest in inscribed 3-polytopes experienced a revival due to their importance in the theory of hyperbolic 3-manifolds [Thu78] and Delaunay triangulations [Bro7... |

1 |
with low-dimensional realization spaces (joint work with K
- Polytopes
(Show Context)
Citation Context ...ension times the total number of its vertices and facets, size(P ) := d ( f0(P ) + fd−1(P ) ) , this problem can be made more concrete as follows. Problem L-S (Legendre–Steinitz in general dimensions =-=[Zie11]-=-). How does, for d-dimensional polytopes, the dimension of the realization space grow with the size of the polytope? We have dimRS(P ) = 12 size(P ) for d = 2 and dimRS(P ) = 13 size(P ) + 4 for d = 3... |