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SHEMER, I. Neighborly polytopes, Israel J. Math. 43 (1982), 291-314. MR 84k:52008 Neighborly polytopes Israel J. Math

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Summary  

The paper gives a collection of open problems on abstract polytopes that were either presented at the \emph{Polytopes Day in Calgary} or motivated by discussions at the preceding \emph{Workshop on Convex and Abstract Polytopes\/} at the Banff International Research Station in May~2005.

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. , The number of vertices of a Fano polytope , Ann. Inst. Fourier (Grenoble) , 56 ( 2006 ), no. 1 , 121 – 130 . [7] Cox , D. , Little

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conjecture and low-dimensional dual cyclic polytopes , Geom. Dedicata , 46 ( 1993 ), no. 3 , 279 – 286 . [3] Bezdek , K. and Bisztriczky

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Abstract  

A convex d-polytope in ℝd is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.

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Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.

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This article announces the creation of an atlas of small regular abstract polytopes. The atlas contains information about all regular abstract polytopes whose automorphism group has order 2000 or less, except those of order \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $512k$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $k\geq1$ \end{document}. The article explains also the techniques used to create the atlas, and gives some summary tables. At the time of printing, the url for the atlas is http://<a href = "http://www.abstract-polytopes.com/atlas">www.abstract-polytopes.com/atlas.

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Let P ⊂ Rn be a centrally symmetric, convex n-polytope with 2r vertices, n ≥ 2. Let P be a family of mn + 1 homothetical copies of P. Based on an algorithmical approach to center hyperplanes of finite point sets in Minkowski spaces with polyhedral norms, we show that a hyperplane transversal of all members of P (if it exists) can be found in O(rm) time when the dimension n is fixed.

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Abstract  

A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d / 2 + 1)d for any d ≥ 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I. Talata [19]. This is a constructive improvement to the result of A. Pr (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices.

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Abstract  

Let V be a finite set of points in the Euclidean d-space (d ≧ 2). The intersection of all unit balls B(υ, 1) centered at υ, where υ ranges over V, henceforth denoted by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(V) is the ball polytope associated with V. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. (a) Define the boundary complex of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(V), i.e., define its vertices, edges and facets in dimension 3, and investigate its basic properties. (b) Apply results of this investigation to characterize finite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950’s by means of ball polytopes, that the diameter of V is attained at most 2|V| − 2 times. Call V extremal if its diameter is attained this maximal number (2|V| − 2) of times. We extend the aforementioned result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(V) and show that in this case the boundary complex of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(V) is self-dual in some strong sense. The problem of constructing new types of extremal configurations is not addressed in this paper, but we do present here some such new types.

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