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Isabel Hubard 《Journal of Combinatorial Theory, Series A》2005,111(1):128-136
We prove that self-dual chiral polytopes of odd rank possess a polarity, that is, an involutory duality, and give an example showing this is not true in even ranks. Properties of the extended groups, that is of the groups of automorphisms and dualities, of self-dual chiral polytopes are discussed in detail. 相似文献
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We show that, for any prime power $n$ and any convex body $K$ (i.e., a compact convex set with interior) in $\mathbb{R }^d$ , there exists a partition of $K$ into $n$ convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature. 相似文献
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We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests
a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to
be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies. 相似文献
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Michael I. Hartley Isabel Hubard Dimitri Leemans 《Journal of Algebraic Combinatorics》2012,35(2):193-214
The Janko group J 1 has, up to duality, exactly two regular rank four polytopes, of respective Schl?fli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph. 相似文献
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Isabel Hubard Alen Orbani? Daniel Pellicer Asia Ivi??Weiss 《Discrete and Computational Geometry》2012,48(4):1110-1136
We derive some general results on the symmetries of equivelar toroids and provide detailed analysis of the subgroup lattice structure of the dihedral group D 4 and of the octahedral group to complete classification by symmetry type of those in ranks 3 and 4. 相似文献
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