Isoperimetric equalities for rosettes

Abstract

In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing 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.

21 pages, 7 figures

References in corpus (4)

• The improved isoperimetric inequality and the Wigner caustic of planar ovals
• A note on Hurwitz's inequality
• Singular points of the Wigner caustic and affine equidistants of planar curves
• Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space