The dual generalized Chernoff inequality for star-shaped curves

Authors: DEYAN ZHANG, YUNLONG YANG

Abstract: In this paper, we first introduce the $k$-order radial function $\rho_k(\theta)$ for star-shaped curves in $\mathbb{R}^2$ and then prove a geometric inequality involving $\rho_k(\theta)$ and the area $A$ enclosed by a star-shaped curve, which can be looked upon as the dual Chernoff--Ou--Pan inequality. As a by-product, we get a new proof of the classical dual isoperimetric inequality. We also prove that $\frac{C^2}{k^2}\leq A<\frac{\pi C^2}{k}$ for star-shaped curves with $\rho_k(\theta)=C (\mathrm{const.})$. In particular, if the curve is equichordal, then $\frac{C^2}{4}\leq A<\frac{\pi C^2}{2}$.

Keywords: Star curves, the dual Chernoff--Ou--Pan inequality, equichordal curves

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