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How many times can the volume of a convex polyhedron be increased by isometric deformations?

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Abstract

We prove that the answer to the question of the title is ‘as many times as you want.’ More precisely, given any constant \(c>0\), we construct two oblique triangular bipyramids, P and Q, in Euclidean 3-space, such that P is convex, Q is nonconvex and intrinsically isometric to P, and \(\text {vol}Q>c\cdot \text {vol}P>0\).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug ave., 4, Novosibirsk, 630090, Russia

    Victor Alexandrov

  2. Department of Physics, Novosibirsk State University, Pirogov st., 2, Novosibirsk, 630090, Russia

    Victor Alexandrov

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  1. Victor Alexandrov
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Correspondence to Victor Alexandrov.

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Alexandrov, V. How many times can the volume of a convex polyhedron be increased by isometric deformations?. Beitr Algebra Geom 58, 549–554 (2017). https://doi.org/10.1007/s13366-017-0336-8

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  • DOI: https://doi.org/10.1007/s13366-017-0336-8

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