Partial inflation of closed polygons in the plane.

*(English)*Zbl 0772.52004Summary: Inflation for simply closed regular curves in the plane has been investigated first by S. A. Robertson [Geometry and topology III, Proc. Workshop, Leeds/UK 1990, 264-275 (1991; Zbl 0727.53004)] and studied in some more detail by S. A. Robertson and the author [to appear in Proc. Conf. Intuitive Geometry, Coll. of the Janos Bolyai Soc.]. It consists of an infinite iteration of reflections of parts of the curve at supporting double tangents, hopefully leading to a convex limit curve which has the same arc length as the original curve. The same procedure easily can be defined for simply closed polygons. It provides a special construction of chord-stretched versions of the given curve.

The aim of this note is to show that the behaviour of inflations is more comfortable in the piecewise linear case. It will end after a finite number of steps. This gives a positive answer to a question posed by T. Kaluza [Math. Semesterber., Neue Folge 28, 153-154 (1981)].

Furthermore inflation may lead to some measure for the nonconvexity of a simply closed polygon.

The aim of this note is to show that the behaviour of inflations is more comfortable in the piecewise linear case. It will end after a finite number of steps. This gives a positive answer to a question posed by T. Kaluza [Math. Semesterber., Neue Folge 28, 153-154 (1981)].

Furthermore inflation may lead to some measure for the nonconvexity of a simply closed polygon.

##### MSC:

52A10 | Convex sets in \(2\) dimensions (including convex curves) |