Convex bodies with equiframed two-dimensional sections
type: Item
instance of: scholarly article
title: Convex bodies with equiframed two-dimensional sections
zbMATH Open document ID: 1113.52011
author: Nico Düvelmeyer
published in: Archiv der Mathematik
publication date: 2007-03-06T00:00:00Z
review text: A convex body in \(\mathbb{R}^n\) is called equiframed if all its boundary points belong to the boundary of a parallelepiped of minimal volume containing that convex body (for \(n=2\) this notion is closely related to that of Radon curves). It is well known that convex bodies in \(\mathbb{R}^n\), \(n\geq 3\), centred at the origin and having the property that any of their sections with a 2-subspace is bounded by a Radon curve, are ellipsoids. The author gives a stronger result: Every convex body in \(\mathbb{R}^n\), \(n\geq 3\), which is centred at the origin and has only equiframed intersections with linear 2-subspaces, is necessarily an ellipsoid. Since Radon curves form a proper subfamily of equiframed curves, this is a nice generalization of a well-known result in classical convexity. Applications in real normed linear spaces are given, too.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A15
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 5131220
zbMATH Keywords: Radon curve
zbMATH Keywords: equiframed curve
zbMATH Keywords: inner product space
zbMATH Keywords: ellipsoid
zbMATH Keywords: Birkhoff orthogonality
zbMATH Keywords: Minkowski plane
zbMATH Keywords: angular measure
zbMATH Keywords: Euclidean space
MaRDI profile type: Publication
full work available at URL: s00013-006-1675-0
OpenAlex ID: W2141623174
DOI: 10.1007/S00013-006-1675-0
description: scientific article; zbMATH DE number 5131220