Dual Orlicz–Brunn–Minkowski theory

Zhu, Baocheng; Zhou, Jiazu; Xu, Wenxue

doi:10.1016/j.aim.2014.07.019

Cited by 105 publications

(68 citation statements)

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“…Recall that there exists a positive such that | | ≥ ‖ ‖ 1 , and ∑ =1 ≥ max 1≤ ≤ = ‖ ‖ = ‖ ‖. Then from the definition of -mean width, Jensen's inequality (Lemma 8, see [30,31]), and (13), it follows that…”

Section: Lemma 11mentioning

confidence: 99%

The Characteristic Properties of the Minimal Lp-Mean Width

2017

Journal of Function Spaces

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Giannopoulos proved that a smooth convex body has minimal mean width position if and only if the measure ℎ ( ) (d ), supported on −1 , is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal -mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal -mean width of convex bodies and prove the existence and uniqueness of the minimal -mean width in its SL( ) images. In addition, we establish a characterization of the minimal -mean width, conclude the average ( ) with a variation of the minimal -mean width position, and give the condition for the minimum position of ( ).

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Section: Lemma 11mentioning

confidence: 99%

The Characteristic Properties of the Minimal Lp-Mean Width

2017

Journal of Function Spaces

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“…Therefore, the equality in Zhu, Zhou and Xu [12]. The dual forms of Theorem 1.1 and Theorem 1.2 were established by Xiong and Zou [11].…”

mentioning

confidence: 97%

“…Zhu, Zhou and Xu [12] defined the Orlicz radial sum and dual Orlicz mixed volumes. Let  was defined by [13] ( )…”

Section: Introductionmentioning

confidence: 99%

Inequalities for Dual Orlicz Mixed Quermassintegrals

Liu¹

2016

APM

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“…The dual OrliczBrunn-Minkowski theory has also attracted mathematicians' attention [14,[26][27][28]. In 2014, Zhu et al [29] introduced the Orlicz harmonic radial sum+ of two star bodies and , defined by…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

Orlicz Mean Dual Affine Quermassintegrals

Zhao

Cheung

2018

Journal of Function Spaces

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Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual OrliczBrunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield the -dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.

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Dual Orlicz–Brunn–Minkowski theory

Cited by 105 publications

References 50 publications

The Characteristic Properties of the Minimal Lp-Mean Width

The Characteristic Properties of the Minimal Lp-Mean Width

Inequalities for Dual Orlicz Mixed Quermassintegrals

Orlicz Mean Dual Affine Quermassintegrals

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