Some remarks about closed convex curves
A well-known result by \textit{P. R. Chernoff} [Am. Math. Mon. 76, 34--35 (1969; Zbl 0175.19501)] states that if \(A\) is the area enclosed by a closed convex curve \(\alpha\) in the plane with width-function \(\omega(\theta)\), then the circle is the only extremal set for the inequality \[ A\leq\frac{1}{2}\int_0^{\pi/2}\omega(\theta)\omega\left(\theta+\tfrac{1}{2}\pi\right)d\theta. \] The width-function satisfies that \(\omega(\theta)=p(\theta)+p(\theta+\pi)\), where \(p\) is the oriented orthogonal distance from a fixed interior point of the region determined by \(\alpha\) to the tangent at a point on \(\alpha\). In this paper the authors extend the notion of width-function, defining, for any integer \(k\geq 2\), \[ \omega_k(\theta)=\sum_{i=0}^{k-1}p\left(\theta+i\,\tfrac{2\pi}{k}\right), \] and show that the circle is the only extremal set for the inequality \[ A\leq\frac{1}{k}\int_0^{\pi/k}\omega_k(\theta)\omega_k\left(\theta+\tfrac{1}{k}\pi\right)d\theta. \] Observe that for \(k=2\), Chernoff's result is obtained. This provides also a new proof of the isoperimetric inequality, since it is proved that the limit of the right hand side in the above expression, when \(k\) goes to infinity, is precisely \(L^2/(4\pi)\), where \(L\) denotes the length of \(\alpha\).
scientific article; zbMATH DE number 5826818