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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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