Stationary partitioning of convex bodies
type: Item
instance of: scholarly article
title: Stationary partitioning of convex bodies
zbMATH Open document ID: 0572.52005
DOI: 10.1007/BF00281743
author: Johannes C. C. Nitsche
published in: Archive for Rational Mechanics and Analysis
publication date: 1985-01-01T00:00:00Z
review text: Let \({\mathcal B}\) be a convex body in \({\mathbb{R}}^ 3\) of volume \(m_ 3({\mathcal B})<\infty\). A disc-type surface \({\mathcal S}\) is spanned across the interior of \({\mathcal B}\) dividing this interior in two simply-connected cells \({\mathcal B}'\) and \({\mathcal B}''\) of volumes \(m_ 3({\mathcal B}')=\sigma m_ 3({\mathcal B}),\quad m_ 3({\mathcal B}'')=(1-\sigma)m_ 3({\mathcal B})\) \((0<\sigma <1)\). The fraction \(\sigma\) is considered as fixed and the problem is to find \({\mathcal S}\) with the smallest possible area or with stationary area. Both conditions imply that the first variation of the area of \({\mathcal S}\) vanishes. A surface with these properties is called stationary. Theorem: For a ball \({\mathcal B}\), all stationary surfaces (which are orientable, simply connected and have Euler characteristic one) are plane discs through the center \((\sigma =1/2)\) and spherical caps orthogonal to \(\partial {\mathcal B}\) (\(\sigma\) \(\neq 1/2).\) Other interesting examples are considered and noteworthy open questions are stated.
Mathematics Subject Classification ID: 52A15
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A99
zbMATH DE Number: 3913146
zbMATH Keywords: minimal surfaces
zbMATH Keywords: mean curvature
zbMATH Keywords: partitioning problem
zbMATH Keywords: convex body
zbMATH Keywords: stationary surfaces
reviewed by: L. A. Santaló
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3913146