Six Topics on Inscribable Polytopes

Padrol, Arnau; Ziegler, Günter M.

doi:10.1007/978-3-662-50447-5_13

Cited by 6 publications

(3 citation statements)

References 42 publications

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“…Indeed, counterexamples to the problem of inscribing convex polytopes of a given combinatorial type in spheres [16] are of this form. For example, the dual to the truncated tetrahedron, known as the triakis tetrahedron, is not inscribable in a sphere; see Figure 4.…”

Section: Convex Geometry Polytopes and Spherical Polytopesmentioning

confidence: 99%

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Malkoun

Olver²

2021

Forum of Mathematics, Sigma

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Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

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Section: Convex Geometry Polytopes and Spherical Polytopesmentioning

confidence: 99%

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Malkoun

Olver²

2021

Forum of Mathematics, Sigma

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“…Earlier, graph‐theoretical necessary conditions and sufficient conditions for a 3‐polytope to be inscribable were provided by Steinitz [26] and [15, Section 13.5] as well as Dillencourt and Smith [8]. Of course, every face of an inscribed polytope must be inscribed, so the inscribability conditions of 3‐polytopes impose natural conditions on higher dimensional polytopes, see, for example, [21, Section 2; 24, Section 12]. In particular, the conditions can be used as a first check to determine the non‐inscribability of some polytopes in dimension 4.…”

Section: Introductionmentioning

confidence: 99%

“…Gonska and Ziegler asked whether inscribable polytopes affect a coarser polytope invariant, the f ‐vector [14, Introduction]. Indeed, experimental results seem to indicate that sufficient conditions for inscribability may be obtained from the f ‐vector [21, Section 2]. For more detail on related questions and their history, we refer to the recent articles [6, 14, 21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Inscribability Obstructions for Higher Dimensional Polytopes

et al. 2020

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For 3-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every f-vector of 3-polytopes, there exists an inscribable polytope with that f-vector. For higher dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the 4-dimensional cyclic polytopes with at least eight vertices-all of whose faces are inscribable-are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic 4-polytopes with up to seven vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that d-dimensional cyclic polytopes with at least d + 4 vertices are not circumscribable, and that no dual of a neighborly 4-polytope with eight vertices, that is, no polytope with f-vector (20,40,28,8), is inscribable.

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Inscribable Order Types

Dobbins,

Lee

2023

Discrete Comput Geom

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Six Topics on Inscribable Polytopes

Cited by 6 publications

References 42 publications

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Combinatorial Inscribability Obstructions for Higher Dimensional Polytopes

Inscribable Order Types

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