  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.

Differential Geometry