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

  • Victor AlexandrovEmail author
Original Paper


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 \(\text {vol}Q>c\cdot \text {vol}P>0\).


Euclidean space Convex polyhedron Bipyramid Intrinsic metric Intrinsic isometry Volume increasing deformation 

Mathematics Subject Classification

52B10 51M20 52A15 52B60 52C25 49Q10 


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Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

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