Volume increasing isometric deformations of convex polyhedra
type: Item
instance of: scholarly article
title: Volume increasing isometric deformations of convex polyhedra
zbMATH Open document ID: 0864.52003
author: David D. Bleecker
published in: Journal of Differential Geometry
publication date: 1996-12-08T00:00:00Z
review text: 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).
reviewed by: Robert Dawson
Mathematics Subject Classification ID: 52A15
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 955318
zbMATH Keywords: volume preserving deformations
zbMATH Keywords: isometric deformations
zbMATH Keywords: convex polyhedra
Wikidata QID: Q115177562
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214458323
OpenAlex ID: W2138724453
DOI: 10.4310/JDG/1214458323
description: scientific article; zbMATH DE number 955318