A reverse isoperimetric inequality for convex plane curves
Let \(\gamma\) be a strictly convex, closed planar curve with length \(L\) and bounding a region with area \(A\). The authors prove the following reverse isoperimetric inequality \[ L^2\leq 4\pi(A+| \Tilde A |), \] where \(\Tilde A\) denotes the oriented area of the domain enclosed by the locus of curvature centers of \(\gamma\), and the equality is attained if and only if \(\gamma\) is a circle. The authors study several interesting properties concerning the locus of the centers of curvature and then obtain the proof integrating the square of the radius of curvature and using both Cauchy's and Blaschke's integral formulas.
locus of centers of curvature, reverse isoperimetric inequality
scientific article; zbMATH DE number 5152757
scientific article