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

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 vol Q> c· vol P> 0.