# A note on Hurwitz's inequality

## TL;DR

This paper refines Hurwitz's inequality for convex plane curves by establishing positive lower bounds for the isoperimetric deficit, involving geometric properties like visual angle and pedal curves, and characterizes equality cases.

## Contribution

It improves Hurwitz's inequality by providing explicit positive lower bounds for the isoperimetric deficit involving geometric invariants and characterizes the equality cases.

## Key findings

- Established lower bounds for the isoperimetric deficit involving visual angle and pedal curves.
- Identified conditions under which equality in the improved inequalities holds.
- Characterized extremal convex sets related to hypocycloids and their Minkowski sums.

## 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 $L^2-4\pi F\leq \pi |F_{e}|$, 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.