scholarly journals Orlicz-Brunn-Minkowski inequality for polar bodies and dual star bodies

2017 ◽  
pp. 1139-1144
Author(s):  
Yan Wang ◽  
Qingzh ng Huang
2019 ◽  
Vol 72 (2) ◽  
pp. 455-479
Author(s):  
Shaoxiong Hou ◽  
Deping Ye

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.


Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2005
Author(s):  
Chang-Jian Zhao

In this paper, the classical dual mixed volume of star bodies V˜(K1,⋯,Kn) and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it Orlicz multiple dual mixed volume. We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of Lp-dual multiple mixed volumes and Lp-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dandan Lai ◽  
Hailin Jin

AbstractThis paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and the equivalent Minkowski inequality for mixed log-volume.


2020 ◽  
Vol 18 (1) ◽  
pp. 1064-1075
Author(s):  
Xia Zhao ◽  
Weidong Wang ◽  
Youjiang Lin

Abstract In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.


2020 ◽  
Vol 18 (1) ◽  
pp. 970-982
Author(s):  
Yibin Feng ◽  
Yanping Zhou

Abstract In this article, the general (p,q)-mixed projection bodies are introduced. Then, some basic properties of the general (p,q)-mixed projection bodies are discussed, and the extreme values of volumes of the general (p,q)-mixed projection bodies and their polar bodies are established.


Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 227 ◽  
Author(s):  
Junjian Zhao ◽  
Wei-Shih Du ◽  
Yasong Chen

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.


2005 ◽  
Vol 53 (3) ◽  
pp. 255-260 ◽  
Author(s):  
Caroline Mackie Ogilvie ◽  
Peter R. Braude ◽  
Paul N. Scriven

Since the early 1990s, preimplantation genetic diagnosis (PGD) has been expanding in scope and applications. Selection of female embryos to avoid X-linked disease was carried out first by polymerase chain reaction, then by fluorescence in situ hybridization (FISH), and an ever-increasing number of tests for monogenic diseases have been developed. Couples with chromosome rearrangements such as Robertsonian and reciprocal translocations form a large referral group for most PGD centers and present a special challenge, due to the large number of genetically unbalanced embryos generated by meiotic segregation. Early protocols used blastomeres biopsied from cleavage-stage embryos; testing of first and second polar bodies is now a routine alternative, and blastocyst biopsy can also be used. More recently, the technology has been harnessed to provide PGD-AS, or aneuploidy screening. FISH probes specific for chromosomes commonly found to be aneuploid in early pregnancy loss are used to test blastomeres for aneuploidy, with the aim of replacing euploid embryos and increasing pregnancy rates in groups of women who have poor IVF success rates. More recent application of PGD to areas such as HLA typing and social sex selection have stoked public controversy and concern, while provoking interesting ethical debates and keeping PGD firmly in the public eye.


2014 ◽  
Vol 104 (3) ◽  
pp. 307-313 ◽  
Author(s):  
Y. Tsutsui ◽  
K. Maeto ◽  
K. Hamaguchi ◽  
Y. Isaki ◽  
Y. Takami ◽  
...  

AbstractAlthough apomixis is the most common form of parthenogenesis in diplodiploid arthropods, it is uncommon in the haplodiploid insect order Hymenoptera. We found a new type of spontaneous apomixis in the Hymenoptera, completely lacking meiosis and the expulsion of polar bodies in egg maturation division, on the thelytokous strain of a parasitoid waspMeteorus pulchricornis(Wesmael) (Braconidae, Euphorinae) on pest lepidopteran larvaeSpodoptera litura(Fabricius) (Noctuidae). The absence of the meiotic process was consistent with a non-segregation pattern in the offspring of heterozygous females, and no positive evidence was obtained for the induction of thelytoky by any bacterial symbionts. We discuss the conditions that enable the occurrence of such rare cases of apomictic thelytoky in the Hymenoptera, suggesting the significance of fixed heterosis caused by hybridization or polyploidization, symbiosis with bacterial agents, and occasional sex. Our finding will encourage further genetic studies on parasitoid wasps to use asexual lines more wisely for biological control.


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