On the girth of convex bodies
type: Item
instance of: scholarly article
title: On the girth of convex bodies
zbMATH Open document ID: 0888.52003
published in: Archiv der Mathematik
publication date: 1998-05-13T00:00:00Z
review text: Let \(K_{E}\) be the orthogonal projection of a convex body \(K\subset\mathbb{R}^{3}\) onto a plane \(E\) through the origin. The perimeter of \(K_{E}\) is called the girth of \(K\) with respect to \(E.\) It is known that the girth of \(K\), as a function of \(E\), does not determine the body \(K\) up to translations, but convex bodies having the same girth are equiwide [see \textit{S. Nakajima}, Jap. J. Math. 7, 225-226 (1930; JFM 56.1204.02)]. For any unit vector \(\mathbf{u}\) in \(E\) that is a regular direction of \(K_{E}\), the ``upper'' arc with respect to \(\mathbf{u}\) is one of the two arcs of \(K_{E}\) with the support points in the direction \(\mathbf{u}\) as endpoints. The length of this ``upper'' arc is called the semi-girth of \(K\) corresponding to \(E\) and \(\mathbf{u}\). The author proves that two convex bodies having the same semi-girth, as a function of \(E\) and \(\mathbf{u}\), are translates of each other. This result is a consequence of a more general stability result on semi-girths. Moreover he establishes an inequality concerning the central symmetry of convex domains.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1052898
zbMATH Keywords: orthogonal projection
zbMATH Keywords: convex body
zbMATH Keywords: girth
zbMATH Keywords: central symmetry
author: Helmut Groemer
reviewed by: Salvatore Vassallo
MaRDI profile type: Publication
full work available at URL: s000130050095
OpenAlex ID: W2006248094
DOI: 10.1007/S000130050095
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description: scientific article; zbMATH DE number 1052898