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

TL;DR

The paper demonstrates that the volume of a convex polyhedron can be increased arbitrarily through isometric deformations by constructing specific examples of convex and nonconvex polyhedra with the same intrinsic geometry.

Contribution

It provides a construction showing that convex polyhedra can be deformed isometrically to arbitrarily larger volumes, answering a longstanding geometric question.

Findings

Volumes can be increased arbitrarily via isometric deformations.

Constructed specific convex and nonconvex polyhedra with equal intrinsic geometry.

Proved the possibility of unbounded volume increase in convex polyhedra.

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 $\textrm{vol} Q>c\cdot \textrm{vol} P>0$.