Survey curves of ovals (Q1129770)

Let \(\Gamma\) be a \(C^1\)-smooth strictly convex closed curve on the Euclidean plane \(E^2\) and let \(\Phi\) be the convex hull of \(\Gamma\). By definition, put \(d(x)=\{ y\in \Gamma | [xy)\cap\Gamma =\emptyset \}\) if \(x\notin\Phi\); \(d(x)=\{ x \}\) if \(x\in\Gamma\); and \(d(x)=\{\emptyset\}\) if \(x\in\Phi\setminus\Gamma\). A path \(L: [0,1]\to E^2\) is called a visibility curve for \(\Gamma\) if \(\cup_{t\in [0,1]} d(L(t))=\Gamma\). It is proven that the following properties of \(L\) are equivalent: (1) \(L\) intersects each straight line intersecting \(\Phi\); (2) \(L\) intersects each tangent line to \(\Gamma\); (3) \(L\) is a visibility curve for \(\Gamma\); (4) \(\Phi\) is contained in the convex hull of \(L\). A simple one-parameter class of visibility curves is constructed. The results obtained are applied to solve the following extremal problems: (P1) find a shortest visibility curve for the unit circle; (P2) given a point \(p\), find a shortest visibility curve \(L\) for the unit circle such that \(p\in L\). Previously problem (P1) was studied by \textit{V.~Faber, J.~Mycielski} and \textit{P.~Pedersen} in [Ann. Polon. Math. 44, 249-266 (1984; Zbl 0561.52018)].