An extension of Rabinowitz's polynomial representation for convex curves
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.
scientific article; zbMATH DE number 7227029