A geometric problem and the Hopf lemma. I (Q2502557)

The authors prove the following result: Let \(M\) be a closed planar \(C^2\)-embedded curve such that \(M\) stays on one side of any tangent to \(M\) parallel to the \(y\)-axis. Under the assumption that for all points \((x, y_1), (x, y_2) \in M\) (points on \(M\) on a line parallel to the \(y\)-axis) with \(y_1 \leq y_2\) the inequality \(\kappa| _{(x, y_2)} \leq \kappa| _{(x, y_1)}\) holds, the curve must be symmetric w.r.t. a line parallel to the \(x\)-axis. (\(\kappa\) denotes the curvature of \(M\).) Some additional results concerning equality of two functions or line-symmetry of their graphs are derived. The paper also outlines some analogous but open problems for higher dimensions. Moreover, counter-examples are given in case of omitting some of the assumptions in the 2-dimensional case.