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How many times can the volume of a convex polyhedron be increased by isometric deformations? (English) Zbl 1387.52024
Summary: 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 \(\operatorname{vol}Q>c\cdot \operatorname{vol}P>0\).
MSC:
52B10 Three-dimensional polytopes
51M20 Polyhedra and polytopes; regular figures, division of spaces
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52B60 Isoperimetric problems for polytopes
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
49Q10 Optimization of shapes other than minimal surfaces
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