Geometric loci associated to certain minimal chords in convex regions (Q1818894)

Among all chords through a fixed point from a strictly convex region \(C\subset\mathbb{R}^2\), the author considers only those having minimal length. Let \(B=bd C\) be the smooth, closed curve bounding \(C\), and let \(P\) denote the set of points from \(C\) with the property that among all chords through \(P\) a minimal one has a prescribed direction \(u\). The author shows that in general the geometric locus of these points is included in two disjoint arcs whose intersection with \(B\) consists of four points two of which determine the tangents of \(C\) parallel to \(u\). He also gives a geometric characterization of the points under consideration belonging to these arcs. It is shown that subarcs, consisting of points close to the two points of tangency, do belong to the geometric locus defined above. Whether the whole geometric locus consists of two (larger) subarcs and, if so, how to construct their other two endpoints, are open problems. Furthermore, the case when \(B\) is an ellipse is discussed, yielding more specified results.