Chord-stretched convex versions of planar curves with self-intersections (Q2707502)

\textbf{Ferenc Fodor (Cookeville)}: Reflecting interior parts of a simply closed curve in the plane at a supporting line and fitting this reflected arc with the untouched part of the curve to a new one, has been considered by several authors as a way to make the curve more convex and to arrive at a convex limit after a possibly infinite number of iterations. For a closed polygon a finite number of iterations will suffice as has been shown by several authors independently. A simply closed \(C^1\)-curve can be convexified after an infinite sequence of iterations unless it has been already convex before. The aim of the paper is to study closed curves with self-intersections. NEWLINENEWLINENEWLINEThe following main statements are proved in the paper. For every closed \(C^1\)-curve \(\gamma\) in the plane there exists a finite number of iterated partial inflations such that after this procedure the resulting curve will be simply closed. Let \(\gamma\) be a closed \(C^1\)-curve in the plane. Then there is a sequence of iterated partial inflations having a convex \(C^1\)-curve as its limit. This is a chord-stretched version of \(\gamma\). NEWLINENEWLINENEWLINEAlso, an outline is given in the case of polygons, indicating how to transfer the arguments used in the proofs of the above statements to closed polygons having self-intersections. The generic case is handled only that the self-intersections happen only in the interior of edges in a transversal way. Let \(\gamma\) be a closed polygon with transversal self-intersections. Then there exists a finite set of iterated partial inflations transforming \(\gamma\) into a simply closed polygon. Moreover, any closed polygon \(\gamma\) with transversal self-intersections can be convexified by a finite number of iterated partial inflations, obtaining a chordstretched version of \(\gamma\). NEWLINENEWLINENEWLINEThe main result of the paper remains valid in spherical geometry, it is a consequence of the fact that the area and intersection estimates used in the proofs can be obtained in the same way as in Euclidean geometry. Thus, if \(\gamma\) is a closed \(C^1\)-curve or closed spherical polygon in the unit sphere and we assume that its length is bounded by \(2\pi\), then there exists a finite number of iterated partial functions of \(\gamma\) transforming \(\gamma\) into a simply closed curve. NEWLINENEWLINENEWLINE\textbf{A.C.Thompson (Halifax)}: In 1935, \textit{P. Erdős} [Problem 3763, Am. Math. Mon. 42, 627 (1935)] posed the following problem: Given a simple closed polygon in the plane can one `convexify'' it by a finite process of reflecting portions of the curve in supporting lines. This was answered affirmatively by \textit{B. de Sz.-Nagy} Am. Math. Mon. 46, 176--177 (1939)]. He gave a simple example to show that the procedure proposed by Erdős might yield curves with self-intersections and proved that if, instead, one uses one line at a time, the process terminates.NEWLINENEWLINE Since then the problem has been considered for simple \(C^1\) curves where an infinite sequence of reflections may be needed to produce a convex curve as a limit. The present paper is concerned with both \(C^1\) curves and polygons that have self-intersections. The author shows that a finite number of steps of Sz.-Nagy's process yields a simple curve to which the earlier results can be applied.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].