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A conjecture on convex polyhedra

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Abstract

We select a class of pyramids of a particular shape and propose a conjecture that precisely these pyramids are of greatest surface area among the closed convex polyhedra having evenly many vertices and the unit geodesic diameter. We describe the geometry of these pyramids. The confirmation of our conjecture will solve the “doubly covered disk” problem of Alexandrov. Through a connection with Reuleaux polygons we prove that on the plane the convex n-gon of unit diameter, for odd n, has greatest area when it is regular, whereas this is not so for even n.

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Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    V. A. Zalgaller

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  1. V. A. Zalgaller
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Correspondence to V. A. Zalgaller.

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To dear Yuriĭ Grigor’evich Reshetnyak on his 80th birthday.

Original Russian Text Copyright © 2009 Zalgaller V. A.

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Rehovot. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 5, pp. 1070–1082, September–October, 2009.

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Zalgaller, V.A. A conjecture on convex polyhedra. Sib Math J 50, 846–855 (2009). https://doi.org/10.1007/s11202-009-0095-3

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  • DOI: https://doi.org/10.1007/s11202-009-0095-3

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