A mixed symmetric Chernoff type inequality and its stability properties
type: Item
instance of: scholarly article
title: A mixed symmetric Chernoff type inequality and its stability properties
zbMATH Open document ID: 1470.52013
author: Deyan Zhang
published in: The Journal of Geometric Analysis
publication date: 2021-05-04T00:00:00Z
review text: 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.
reviewed by: Judit Abardia
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 7343452
zbMATH Keywords: Chernoff type inequality
zbMATH Keywords: \(k\)-order width
zbMATH Keywords: Wigner caustic
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: s12220-020-00485-0
OpenAlex ID: W3044501643
cites work: Stability of the Blaschke-Santaló and the affine isoperimetric inequality
cites work: Q4039749
cites work: An Area-Width Inequality for Convex Curves
cites work: A note on Hurwitz's inequality
cites work: A sharp stability result for the relative isoperimetric inequality inside convex cones
cites work: A mass transportation approach to quantitative isoperimetric inequalities
cites work: Stability in the Isoperimetric Problem
cites work: Q5699144
cites work: The sharp quantitative isoperimetric inequality
cites work: Stability properties of the generalized Chernoff inequality
cites work: The Brunn-Minkowski inequality
cites work: Stability Theorems for Convex Domains of Constant Width
cites work: Stability Properties of Geometric Inequalities
cites work: Stability theorems for two measures of symmetry
cites work: Stability Estimates for some Geometric Inequalities
cites work: A sharp lower bound on the polygonal isoperimetric deficit
cites work: On the stability of the polygonal isoperimetric inequality
cites work: Mixed width-integrals of convex bodies
cites work: The isoperimetric inequality
cites work: Bonnesen-Style Isoperimetric Inequalities
cites work: Some remarks about closed convex curves
cites work: Stability of a reverse isoperimetric inequality
cites work: A reverse isoperimetric inequality for convex plane curves
cites work: A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature
cites work: Convex Bodies The Brunn-MinkowskiTheory
cites work: 平面两凸域的Bonnesen型对称混合不等式
cites work: Bonnesen-Style Symmetric Mixed Isoperimetric Inequality
cites work: Q4900488
cites work: Inequalities with curvature and their stability estimates for convex curves
cites work: The improved isoperimetric inequality and the Wigner caustic of planar ovals
DOI: 10.1007/S12220-020-00485-0
description: scientific article; zbMATH DE number 7343452