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.2010 Mathematics Subject Classification. Primary: 52A38, 53A04, 58K70. Secondary: 52A40.

The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

“…Let m be even and k m or m be odd and k < m. By Theorem 2.9 in [43] we know that E 0.5,k (R m ) has at least 2 cusp singularities. Because the cusp in E 0.5 appears when κ(a) κ(b) = 1 and cusp in CSS appears when…”

“…A hedgehog can be also viewed as the Minkowski difference of convex bodies (see [23,24,25,26,27]). The non-singular hedgehogs are also known as the rosettes (see [2,30,43]).…”

“…This construction can be generalized to higher even dimensions ( [4]). The oriented area of the Wigner caustic improves the classical planar isoperimetric inequality and gives the relation between the area and the perimeter of smooth convex bodies of constant width ( [41,42,43]).…”

“…We will study the geometry of E(R m ) through the support function of R m ( [2,43]). Take a point O as the origin of our frame.…”

“…Remark 5.6. In [9,43] we study in details the geometry of affine λ-equidistants of rosettes. We show among other things that there exist m branches of E 0.5 (R m ) and 2m − 1 branches of E λ (R m ) for λ = 0, 0.5, 1.…”