A note on Hurwitz's inequality

Abstract

Given a simple closed plane curve $\Gamma$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4\pi F\le \pi |F_e|$, where $F_e$ is the algebraic area enclosed by the evolute of $\Gamma$. In this note we improve this inequality finding strictly positive lower bounds for the deficit $\pi |F_e|-\Delta$, where $\Delta =L^2-4\pi F$. These bounds involve wether the visual angle of $\Gamma$ or the pedal curve associated to $K$ with respect to the Steiner point of $K$ or the ${\mathcal{L}}^2$ distance between $K$ and the Steiner disk of $K$. For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of $3,4$ or $5$ cusps or the Minkowski sum of this kind of sets.

15 pages, 3 figures

References in corpus (2)

• The improved isoperimetric inequality and the Wigner caustic of planar ovals
• A lower bound for the isoperimetric deficit

Cited by in corpus (3)

• Singular points of the Wigner caustic and affine equidistants of planar curves
• Isoperimetric equalities for rosettes
• The Geometry of the Wigner Caustic and a Decomposition of a Curve Into Parallel Arcs