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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 aregion of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4\piA_{M}. \end{align*} In this paper we present the improved isoperimetricinequality, which states that if $M$ is a closed regular simple convex curve,then \begin{align*} L_{M}^2\geqslant 4\piA_{M}+8\pi\left|\widetilde{A}_{E_{\frac{1}{2}}(M)}\right|, \end{align*} where$\widetilde{A}_{E_{\frac{1}{2}}(M)}$ is an oriented area of the Wigner causticof $M$, 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 isoperimetricinequality (near equality implies curve nearly of constant width). The Wignercaustic is an example of an affine $\lambda$-equidistant (for$\displaystyle\lambda=\frac{1}{2}$) and the improved isoperimetric inequalityis a consequence of certain bounds of oriented areas of affine equidistants.

The improved isoperimetric inequality and the Wigner caustic of planar ovals