Eigenvalue equations in curve theory. I: Characterizations of conic sections (Q5954158)

Let \(c:[0,2\pi] \to \mathbb{R}^2\) be an oval in the Euclidean plane with parameter \(\sigma\), equal to the arc length of the spherical image. The oval \(c\) is an ellipse if and only if the curvature \(k\) satisfies the equation \({d^2\over ds^2} (k^{2/3}- \overline k)+4 (k^{2/3}- \overline k)=0\), where \(\overline k\) is the mean value of \(k^{2/3}\). This global result follows from a local characterization of conic sections by a similar equation. Similar results have been proved by using the support function instead of curvature.