The improved isoperimetric inequality and the Wigner caustic of planar ovals

The classical isoperimetric inequality in the Euclidean plane ${\mathbb{R}}^2$ states that for a simple closed curve $M$ of the length $L_M$, enclosing a region of the area $A_M$, one gets \begin{align*} L_{M}^2\geqslant 4\pi A_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which states that if $M$ is a closed regular simple convex curve, then \begin{align*} L_{M}^2\geqslant 4\pi A_{M}+8\pi\left|\widetilde{A}{E{\frac{1}{2}}(M)}\right|, \end{align*} where ${\tilde{A}}_{E_{\frac{1}{2}}(M)}$ is an oriented area of the Wigner caustic of $M$, and the equality holds if and only if $M$ is a curve of constant width. Furthermore we also present a stability property of the improved isoperimetric inequality (near equality implies curve nearly of constant width). The Wigner caustic is an example of an affine $\lambda$-equidistant (for ${\displaystyle \lambda =\frac{1}{2}}$) and the improved isoperimetric inequality is a consequence of certain bounds of oriented areas of affine equidistants.