Volume increasing isometric deformations of convex polyhedra

In this paper, we prove that the surface of a generic convex polyhedron in E 3 can be isometrically deformed so as to enclose greater volume.Our deformed surfaces have at least seven times more faces than the original surface.They are extrinsically nonconvex, but have the same intrinsic geometry (e.g., area, angle deficits, etc.).Roughly put, the deformations are obtained by simultaneously delivering karate chops to the edges of the polyhedron.While the edges implode, to preserve the intrinsic geometry, portions of the faces move outward, leading to a net increase of volume.The regular tetrahedron can be isometrically deformed to enclose over 37.7% more volume, while the cube, octahedron, dodecahedron, and icosahedron enjoy increases over 21.8%, 11.5%, 9.3%, and 3.6%, respectively.Many questions have arisen.In particular, what is the nature of an isometric embedding, if any, of an abstract surface which encloses the maximum volume?We propose that the term "sandbag" be used to describe such an embedding.What kinds of singularities can they have?Are they unique up to Euclidean motions?