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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 1, pp. 243-259 (2002)
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On Perfect 4-Polytopes

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Gabor Gévay

Department of Geometry, University of Szeged, Aradi vértanuk tere 1, H-6720 Szeged, Hungary, e-mail: gevay@math.u-szeged.hu

**Abstract:** The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope $P$ is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group $G(P)$ on its face-lattice $F(P)$. By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper 2 new classes of perfect 4-polytopes are constructed and discussed, hence Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet.

**Keywords:** perfect polytope; Rostami conjecture; Wythoff polytopes; classification of perfect 4-polytopes

**Classification (MSC2000):** 52B15

**Full text of the article:**

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