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

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 $Q$, such that $P$ is convex, $Q$ is nonconvex and intrinsically isometric to $P$, and $\textrm{vol} Q>c\cdot \textrm{vol} P>0$.