Inflation of plane curves.

*(English)*Zbl 0727.53004
Geometry and topology III, Proc. Workshop, Leeds/UK 1990, 264-275 (1991).

[For the entire collection see Zbl 0724.00015.]

According to the author inflation is as a geometrical construction which assigns to any \(C^ 1\)-embedding f: \(S^ 1\to E^ 2\) a \(C^ 1\)- immersion \(\phi (f):S^ 1\to E^ 2\) of the same length. It is obtained by reflecting all parts of the curve which do not belong to the boundary of the convex hull of its image at the corresponding supporting tangents of this convex hull. If \(\phi\) (f) is an embedding again then the construction may be repeated, and if this can be done infinitely many times then f is called fully inflatable. In this case the author shows that the sequence \(\phi^ n(f)\), \(n\in {\mathbb{N}}\), converges in a \(C^ 1\)-sense to a convex \(C^ 1\)-curve having same length as f.

Though it is not hard to prove that there is a big class of nonconvex \(C^ 1\)-curves which are fully inflatable, it is rather difficult to verify this property in explicit cases. The author poses the question if a circle can occur as a limit curve under full inflation. This has been answered positively in a subsequent paper by S. A. Robertson and B. Wegner [Full and partial inflation of plane curves (to appear)].

According to the author inflation is as a geometrical construction which assigns to any \(C^ 1\)-embedding f: \(S^ 1\to E^ 2\) a \(C^ 1\)- immersion \(\phi (f):S^ 1\to E^ 2\) of the same length. It is obtained by reflecting all parts of the curve which do not belong to the boundary of the convex hull of its image at the corresponding supporting tangents of this convex hull. If \(\phi\) (f) is an embedding again then the construction may be repeated, and if this can be done infinitely many times then f is called fully inflatable. In this case the author shows that the sequence \(\phi^ n(f)\), \(n\in {\mathbb{N}}\), converges in a \(C^ 1\)-sense to a convex \(C^ 1\)-curve having same length as f.

Though it is not hard to prove that there is a big class of nonconvex \(C^ 1\)-curves which are fully inflatable, it is rather difficult to verify this property in explicit cases. The author poses the question if a circle can occur as a limit curve under full inflation. This has been answered positively in a subsequent paper by S. A. Robertson and B. Wegner [Full and partial inflation of plane curves (to appear)].

Reviewer: Bernd Wegner (Berlin)