Extremal convex planar sets
type: Item
instance of: scholarly article
title: Extremal convex planar sets
zbMATH Open document ID: 1068.52009
author: Lu Ting
author: Joseph B. Keller
published in: Discrete \& Computational Geometry
publication date: 2005-04-28T00:00:00Z
review text: The authors obtain several inequalities relating the area, the perimeter, the diameter and the minimal width of a planar convex set; they also determine the extremal sets. They try to build a complete system of inequalities using also some classical inequalities proved by Lebesgue, Kubota Fukasawa, Sholander and Hemmi. A system of inequalities concerning \(n\) geometric magnitudes of a set is said to be complete if for any collection of \(n\) numbers satisfiying those conditions, a set with these values of the characteristics exists. This notion was introduced by Blaschke. From a complete system of inequalities all inequalities concerning these magnitudes can be found. Since Blaschke several complete systems concerning three parameters have been obtained (although Blaschke's original problem is still open). This paper represents the first attempt to build a complete system of inequalities concerning four parameters. They provide an alternative version of Blaschke's diagram as the cross-section of a cone; this diagram turns out to be a compact set in 3-dimensional euclidean space whose boundary is a finite family of surfaces each one corresponding to one inequality. The authors succeed in determining nearly all the surfaces; only the bottom boundary is missing, and they present an interesting conjecture for the bottom boundary.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 2162289
zbMATH Keywords: complete systems of inequalities
zbMATH Keywords: geometric inequalities
zbMATH Keywords: area
zbMATH Keywords: perimeter
zbMATH Keywords: diameter
zbMATH Keywords: minimal width
MaRDI profile type: Publication
full work available at URL: s00454-004-1145-z
OpenAlex ID: W1967988708
DOI: 10.1007/S00454-004-1145-Z
description: scientific article; zbMATH DE number 2162289