A mixed symmetric Chernoff type inequality and its stability properties
Let \(\alpha\) be a convex curve in the Euclidean plane. The classical Chernoff inequality states that \[A\leq \frac{1}{2}\int_{0}^{\pi/2}w(\theta)w\left(\theta+\frac{\pi}{2}\right)d\theta,\] where \(A\) denotes the area and \(w\) the width function of \(\alpha\). Equality holds if and only if \(\alpha\) is a circle. For \(k\geq 2\), the \(k\)-order width function \(w_k\) is defined by \[w_k(\theta)=h(\theta)+\dots+h\left(\theta+\frac{2(k-1)\pi}{k}\right),\quad \theta\in[0,2\pi)\] where \(h\) denotes the support function of \(\alpha\). The author proves the following mixed Chernoff type inequality for two convex domains \(P\) and \(Q\): \[\sqrt{A_PA_Q}\leq \frac{1}{2k}\int_{0}^{\frac{2\pi}{k}}w_k(P,\theta)w_k(Q,\theta+\frac{\pi}{k})d\theta,\] with equality if and only if \(P\) and \(Q\) are circles. A related inequality involving the Wigner caustic of a convex body and a stability result for the obtained mixed symmetric Chernoff type inequality are also proved in the paper.
scientific article; zbMATH DE number 7343452
scientific article