A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature
type: Item
instance of: scholarly article
title: A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature
zbMATH Open document ID: 0909.53002
author: Ralph Howard
author: Andrejs Treibergs
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-04-08T00:00:00Z
review text: The authors study planar imbedded (nonconvex) closed curves \(\gamma\) whose geodesic curvature \(K_g\) satisfies the bound \(| K_g|\leq 1\). The main result can be reformulated as follows. Let \(A\) denote the area of the region bounded by \(\gamma\). Then there is a constant \(C>0\) with the following properties: (i) If \(A\leq C\), then there is a positive increasing function \(L(A)\) such that \(\text{length}(\gamma)\leq L(A)\), with equality exactly for a 1-parameter family \(P_\delta\) of peanut-shaped regions, where \(\delta=4\sin \left({L(A)- 4\pi\over 8}\right)\). (ii) If \(A>C\), then \(\gamma\) can be arbitrarily long compared to \(A\). Here \(C= \pi+2\sqrt 3\) and \(L(A)=2\pi+4\text{arcsin} \left({A-\pi\over 4} \right)\). The results depend on the theorem of \textit{G. Pestov} and \textit{V. Ionin} [Dokl. Akad. Nauk SSSR 127, 1170-1172 (1959; Zbl 0086.36104)] on, in modern terminology, the filling radius of a planar imbedded curve of bounded curvatue [cf. \textit{M. Gromov}, J. Differ. Geom. 18, 1-147 (1983; Zbl 0515.53037)].
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 796894
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: cut point
zbMATH Keywords: focal point
zbMATH Keywords: closed plane curve
zbMATH Keywords: geodesic curvature
zbMATH Keywords: filling radius
reviewed by: Mikhail G. Katz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072242
OpenAlex ID: W2055479476
cites work: Q3876608
cites work: Q4067705
cites work: Q4733008
cites work: Q4023644
cites work: Q4091004
cites work: On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents
cites work: Q5725219
cites work: Regularity of minimizing surfaces of prescribed mean curvature
cites work: On boundary branch points of minimizing surfaces
cites work: Die erweiterten Steinerschen Formeln für ebene und sphärische Bereiche
cites work: Q4733040
cites work: Q3259414
cites work: Q5182583
cites work: A proof of the regularity everywhere of the classical solution to Plateau's problem
cites work: Q3259413
cites work: Variational methods in optimum control theory
cites work: Q3283426
DOI: 10.1216/RMJM/1181072242
description: scientific article; zbMATH DE number 796894