Isoperimetric equalities for rosettes.

In this paper, we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with nonvanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set. We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes. [ABSTRACT FROM AUTHOR]