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In this paper we introduce and study sets: the Constant Width Measure Set andthe Spherical Measure Set, which measure the constant width property and thespherical property of planar regular simple curves, respectively.  Using these sets and the Wigner caustic, we find the exact relations betweenthe length and the area bounded by a closed regular simple convex planar curve$M$. Namely, the following equalities are fulfilled: \begin{align}\label{AbstractEq1} L_{M}^2 &=4\piA_M+8\pi\left|\widetilde{A}_{E_{\frac{1}{2}}(M)}\right|+\pi\left|\widetilde{A}_{CWMS(M)}\right|,\\\label{AbstractEq2} L_{M}^2 &=4\pi A_M+4\pi\left|\widetilde{A}_{SMS(M)}\right|,\end{align} where $L_{M}, A_{M}, \widetilde{A}_{E_{\frac{1}{2}}(M)},\widetilde{A}_{CWMS(M)}, \widetilde{A}_{SMS(M)}$ are the length of $M$, thearea bounded by $M$, the oriented area of the Wigner caustic of $M$, theoriented area of the Constant Width Measure Set of $M$ and the oriented area ofthe Spherical Measure Set of $M$, respectively. The second isoperimetricequality holds also if $M$ is a regular closed simple curve.  We also present a new proof of Four-Vertex Theorem for generic ovals based onthe number of singular points of the $SMS$.

The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals