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

Abstract

In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between the length and the area of the region bounded by an oval $M$. Namely, the following equality is fulfilled: \begin{align*} L_{M}^2 &=4\pi A_M+8\pi\left|\widetilde{A}_{\mathrm{E}_{\frac{1}{2}}(M)}\right|+\pi\left|\widetilde{A}_{\mathrm{CWMS}(M)}\right|, \end{align*} where $L_{M}, A_{M}, \widetilde{A}_{\mathrm{E}_{\frac{1}{2}}(M)}, \widetilde{A}_{\mathrm{CWMS}(M)}$ are the length of $M$, the area bounded by $M$, the oriented area of the Wigner caustic of $M$ and the oriented area of the Constant Width Measure Set of $M$, respectively. Furthermore we study the geometry of the Spherical Measure Set, which is an offset of a curve with a special distance. We show that the oriented area of this set of an oval $M$, $\widetilde{A}_{\mathrm{SMS}(M)}$, satisfies the following equality: \begin{align*} 4\left|\widetilde{A}_{\mathrm{SMS}(M)}\right|=8\left|\widetilde{A}_{\mathrm{E}_{\frac{1}{2}}(M)}\right|+\left|\widetilde{A}_{\mathrm{CWMS}(M)}\right|. \end{align*}