A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature
The authors study planar imbedded (nonconvex) closed curves \(\gamma\) whose geodesic curvature \(K_g\) satisfies the bound \(| K_g|\leq 1\). The main result can be reformulated as follows. Let \(A\) denote the area of the region bounded by \(\gamma\). Then there is a constant \(C>0\) with the following properties: (i) If \(A\leq C\), then there is a positive increasing function \(L(A)\) such that \(\text{length}(\gamma)\leq L(A)\), with equality exactly for a 1-parameter family \(P_\delta\) of peanut-shaped regions, where \(\delta=4\sin \left({L(A)- 4\pi\over 8}\right)\). (ii) If \(A>C\), then \(\gamma\) can be arbitrarily long compared to \(A\). Here \(C= \pi+2\sqrt 3\) and \(L(A)=2\pi+4\text{arcsin} \left({A-\pi\over 4} \right)\). The results depend on the theorem of \textit{G. Pestov} and \textit{V. Ionin} [Dokl. Akad. Nauk SSSR 127, 1170-1172 (1959; Zbl 0086.36104)] on, in modern terminology, the filling radius of a planar imbedded curve of bounded curvatue [cf. \textit{M. Gromov}, J. Differ. Geom. 18, 1-147 (1983; Zbl 0515.53037)].
scientific article; zbMATH DE number 796894
scientific article