An extension of Rabinowitz's polynomial representation for convex curves
type: Item
instance of: scholarly article
title: An extension of Rabinowitz's polynomial representation for convex curves
zbMATH Open document ID: 1453.53005
author: Laiyuan Gao
author: Zhongyun Zhang
author: Fei Zhou
published in: Beiträge zur Algebra und Geometrie
publication date: 2020-08-03T00:00:00Z
review text: Consider a smooth, closed and embedded plane curve \(C\) with positive curvature \(\kappa\) everywhere, that means \(C\) is a convex curve. Denoting by \(s\) the arclength, \(L\) the length and \(A\) the bounded area, an inequality due to \textit{Y.-C. Lin} and \textit{D.-H. Tsai} [J. Evol. Equ. 12, No. 4, 833--854 (2012; Zbl 1266.53062)] states \[ \int_{C}\frac{1}{\kappa}ds\geq 2A+\frac{2(L^2-4\pi A)}{\pi}. \] Using a parametrization \(X(\varphi):S^1\rightarrow {\mathbb R}^2\) of \(C\), Lin and Tsai proved that equality above holds iff the support function \(p:=-\langle X,N\rangle\), \(N\) the unit normal vector, has the form \[ p(\Theta)=\frac{L}{2 \pi}+a_1 \cos\Theta+ b_1 \sin\Theta+a_2 \cos(2 \Theta)+b_2 \sin(2 \Theta),\] with \(\Theta\) the tangent angle and \(a_1, b_1, a_2, b_2\) are constants. Using a method of \textit{S. Rabinowitz} [Missouri J. Math. Sci. 9, No. 1, 23--27 (1997; Zbl 1097.52501)], the authors of the present paper prove at first that the convex curves with the above support functions are polynomial curves of degree at most 6. The authors construct more convex algebraic curves by extending Rabinowitz's method. Especially, they give examples of algebraic convex curves of constant width.
reviewed by: Friedrich Manhart
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 7227029
zbMATH Keywords: resultant
zbMATH Keywords: constant width
zbMATH Keywords: polynomial curve
MaRDI profile type: Publication
full work available at URL: s13366-020-00494-8
OpenAlex ID: W3009568012
cites work: Curves of Constant Width from a Linear Viewpoint
cites work: Positive Centers and the Bonnesen Inequality
cites work: Evolving convex curves to constant-width ones by a perimeter-preserving flow
cites work: On Yau's problem of evolving one curve to another: convex case
cites work: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves
cites work: Evolving a convex closed curve to another one via a length-preserving linear flow
cites work: Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves
cites work: Bodies of Constant Width
cites work: On a non-local perimeter-preserving curve evolution problem for convex plane curves
cites work: A polynomial curve of constant width
cites work: Smooth curves of constant width
DOI: 10.1007/S13366-020-00494-8
description: scientific article; zbMATH DE number 7227029