Volume increasing isometric deformations of convex polyhedra
The title of this article may cause a double-take among readers, familiar with the results of Cauchy and Aleksandrov, who remember that within the class of convex surfaces there are no volume-increasing isometric deformations. However, the deformations used here `crush in' the centers of the edges of a polyhedron, and stretch the corners out into blunt spikes. (The article makes this clearer by abundant and well-drawn illustrations.) Any polyhedron may be deformed in this way, and the result is always nonconvex. Moreover, the author shows that many polyhedra (including the regular ones) may have their volume increased by a carfully-chosen such deformation. In particular, he proves such a result for all convex polyhedra with triangular faces. He also obtains the maximum volumes of polyhedra obtained in this way from regular polyhedra and closes with an assortment of open problems of a similar nature. In particular, he introduces the concept of a `sandbag': a body with maximal volume among those whose surfaces are in a given isometric equivalence class. Surprisingly, even some ellipsoids are not sandbags; indeed, it seems not to be known now whether any convex body other than a sphere has this property. (A typical nonsandbag may be `inflated', like a Mylar balloon, at the cost of nonconvex puckers in some places).
scientific article; zbMATH DE number 955318