Bonnesen-Style Symmetric Mixed Isoperimetric Inequality

Xu, Wenxue; Zhou, Jiazu; Zhu, Baocheng

doi:10.1007/978-4-431-55215-4_9

Cited by 5 publications

(3 citation statements)

References 31 publications

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“…By comparing a simple closed curve and a circle, Schmidt proved the isoperimetric inequality in 1938. We were motivated by Schmidt's works, we compared the two simple closed curves directly and obtained the symmetric mixed isoperimetric inequality (see [14,20,21,[24][25][26]30]). That is, let K k (k = 0, 1) be two domains of areas A k with simple boundaries of perimeters P k in R 2 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…An inequality of the form (1.9) is called the Bonnesen-style symmetric mixed inequality, it is stronger than the symmetric mixed isoperimetric inequality (1.7). Zhou, Xu, Zeng, and others (see [14,20,21,[24][25][26]30]) obtained some Bonnesen-style symmetric mixed inequalities with the known kinematic formulae of Poincaré and Blaschke. Conversely, we considered the upper bound of the symmetric mixed isoperimetric deficit of K 0 and K 1 , that is,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Let n{∂K 0 ∩ ∂(t(gK 1 ))} denote the number of points of intersection ∂K 0 ∩ ∂(t(gK 1 )), and let χ{K 0 ∩ t(gK 1 )} be the Euler-Poincaré characteristics of the intersection K 0 ∩ t(gK 1 ). Then we have the following kinematic formula of Poincaré (see [14,20,21,[24][25][26]30]):…”

Section: Reverse Bonnesen-style Symmetric Mixed Inequalitiesmentioning

confidence: 99%

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A sharp reverse Bonnesen-style inequality and generalization

Wang

2019

J Inequal Appl

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We investigate the isoperimetric deficit of the oval domain in the Euclidean plane. Via the kinematic formulae of Poincaré and Blaschke, and Blaschke's rolling theorem, we obtain a sharp reverse Bonnesen-style inequality for a plane oval domain, which improves Bottema's result. Furthermore, we extend the isoperimetric deficit to the symmetric mixed isoperimetric deficit for two plane oval domains, and we obtain two reverse Bonnesen-style symmetric mixed inequalities, which are generalizations of Bottema's result and its strengthened form.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A sharp reverse Bonnesen-style inequality and generalization

Wang

2019

J Inequal Appl

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Translative containment measure and symmetric mixed isohomothetic inequalities

Luo

Zhou

2015

Sci. China Math.

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We first investigate the translative containment measure for convex domain K 0 to contain, or to be contained in, the homothetic copy of another convex domain K 1 , i.e., given two convex domains K 0 , K 1 of areas A 0 , A 1 , respectively, in the Euclidean plane R 2 , is there a translation T so that t(T K 1 ) ⊂ K 0 or t(T K 1 ) ⊃ K 0 for t > 0? Via the translative kinematic formulas of Poincaré and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficitis the mixed area of K 0 and K 1 . We obtain a sufficient condition for K 0 to contain, or to be contained in, t(T K 1 ). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality. Keywordstranslative containment measure, symmetric mixed isohomothetic deficit, symmetric mixed isohomothetic inequality, Bonnesen-style symmetric mixed isohomothetic inequality, reverse Bonnesen-style symmetric mixed isohomothetic inequality MSC(2010) 52A10, 52A22Citation: Luo M, Xu W X, Zhou J Z. Translative containment measure and symmetric mixed isohomothetic inequalities.

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A Mixed Symmetric Chernoff Type Inequality and Its Stability Properties

Zhang

2020

J Geom Anal

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Bonnesen-Style Symmetric Mixed Isoperimetric Inequality

Cited by 5 publications

References 31 publications

A sharp reverse Bonnesen-style inequality and generalization

A sharp reverse Bonnesen-style inequality and generalization

Translative containment measure and symmetric mixed isohomothetic inequalities

A Mixed Symmetric Chernoff Type Inequality and Its Stability Properties

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