On the minimal convex annulus of a planar convex body
type: Item
instance of: scholarly article
title: On the minimal convex annulus of a planar convex body
zbMATH Open document ID: 0765.52003
DOI: 10.1007/BF01535579
author: C. Peri
author: Andreana Zucco
published in: Monatshefte für Mathematik
publication date: 1993-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/178581
review text: Let \(K\) be a fixed smooth and strictly convex body in \(E^ 2\) with 0 in its interior. By a convex annulus with centre \(c\) we mean the set of all points inside the convex body of the form \(\rho K+c\) and outside of the form \(\sigma K+c\), where \(0\leq\sigma\leq\rho\). A minimal convex annulus \({\mathcal K}(C)\) of a convex body \(C\) is a convex annulus containing the boundary of \(C\) and for which \(|\rho-\sigma|\) is minimal. We first generalize results of Bonnesen, Kritikos and Barany on ``circular'' annulus by proving the existence of a Radon partition of the set of contact points of the boundary of \({\mathcal K}(C)\) and \(C\). Subsequently the uniqueness of \({\mathcal K}(C)\) is shown. Finally we extend a result of the second author on circular annulus showing that for a typical convex body \(C\) (typical in the sense of Baire categories) the minimal convex annulus meets the boundary of \(C\) in precisely four points.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 164918
zbMATH Keywords: Baire category
zbMATH Keywords: minimal convex annulus
zbMATH Keywords: convex body
zbMATH Keywords: Radon partition
MaRDI profile type: Publication
cites work: Q3797805
cites work: Q3974575
description: scientific article; zbMATH DE number 164918