The geometry of wide curves in the plane (Q1039871)

The article characterizes the shortest planar curves of minimal width \(\Delta\) joining two points in the plane. Without loss of generality, it is assumed that the coordinates of the end points of the curve are \((\frac{\delta}{2}, 0)\) and \((-\frac{\delta}{2}, 0)\); it is also assumed that there is at least one point in the image of the curve in the open upper half-plane. Such curves are called \textit{based} curves. For \(\delta\geq 0\) the set \(H_{\delta}\) is defined as \[ H_{\delta}=Conv(\mathbb{S}^1 \cup\{(\pm \delta,0), (\pm \frac{\delta}{2},1),(\pm \frac{\delta}{2},-1)\}). \] The main result states that a based curve \(\gamma\) is the shortest planar curve of minimal width \(\Delta\) joining \((\frac{\delta}{2}, 0)\) and \((-\frac{\delta}{2}, 0)\) if and only if \(\gamma\) is a convex curve and \[ Conv(\gamma)+(-Conv(\gamma))=\Delta H_{\frac{\delta}{\Delta}}. \] It is also proved that the solution curve is unique if \(\delta \geq \Delta\) and that there are infinitely many solutions if \(\delta < \Delta\). Besides the author proves that the width function of the solution curve is unique and he provides explicit formulae for the perimeter of \(H_{\delta}\) and for the length of any solution curve.