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Axioms
Special Issues
Advances in Convex Geometry and Analysis
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Advances in Convex Geometry and Analysis

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: 29 November 2024 | Viewed by 3881

Special Issue Editors

Prof. Dr. Baocheng Zhu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, China
Interests: convex geometric analysis; integral geometry; geometric inequalities
Prof. Dr. Senlin Wu

E-Mail Website
Guest Editor
School of Mathematics, North University of China, Taiyuan 030051, China
Interests: Banach space theory; convex and discrete geometry
Prof. Dr. Jingbo Dou

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, China
Interests: nonlinear elliptic equation; integral inequality; extremal function; variational methods
Dr. Wenxue Xu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Southwest University, Chongqing, China
Interests: convex geometric analysis; integral geometry

Special Issue Information

Dear Colleagues,

Convex geometric analysis is the subject that studies geometric structures and invariants of convex sets using both geometric and analytic methods. Results from the convex geometric analysis have been applied in numerous mathematical disciplines: stochastic geometry, integral geometry, differential geometry, Minkowski and Finsler geometry, combinatorial geometry, algebraic geometry, non-linear partial differential equations, especially the Monge–Ampere equations, number theory, Banach space theory, probability and multivariate statistics.

The aim of this Special Issue is to collate original and high-quality research and review articles related to the development of and applications in convex geometry and analysis. We also hope to attract review articles which describe the current state of the art within this field.

Potential topics include, but are not limited to, the following:

  • Geometric inequalities, isoperimetric inequalities;
  • Minkowski type problem;
  • Differential and integral equations;
  • The completeness of weighted Lp spaces;
  • Banach space theory;
  • Differential geometry;
  • Discrete geometry.

We look forward to receiving your contributions.

Prof. Dr. Baocheng Zhu
Prof. Dr. Senlin Wu
Prof. Dr. Jingbo Dou
Dr. Wenxue Xu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Borsuk’s problem
  • Hadwiger’s covering problem
  • complete sets
  • geometric inequality
  • Brunn-Minkowski inequality
  • Minkowski problem
  • integral inequality
  • Weighted Lp spaces
  • holder’s inequality
  • Fourier coefficients

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

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Research

14 pages, 504 KiB  
Article
Spherical Steiner Symmetrizations
by Youjiang Lin and Zhilang Deng
Axioms 2024, 13(11), 751; https://doi.org/10.3390/axioms13110751 (registering DOI) - 31 Oct 2024
Viewed by 111
Abstract
In this paper, we primarily investigate and establish several properties of spherical Steiner symmetrizations, along with the isoperimetric property of the spherical cap in Sn. Specifically, we study the monotonically decreasing property of the measure of the symmetric difference of two [...] Read more.
In this paper, we primarily investigate and establish several properties of spherical Steiner symmetrizations, along with the isoperimetric property of the spherical cap in Sn. Specifically, we study the monotonically decreasing property of the measure of the symmetric difference of two spherical compact sets, the monotonically decreasing property of the spherical diameter of a spherical compact set, the convergence of iterative spherical Steiner symmetrizations, and so on. In particular, we prove that the sequence of iterative spherical Steiner symmetrizations of K⊂Sn, which follow sequences selected from a finite set of directions, converges to a spherical cap with the same measure as K, extending the result from Rn to Sn on Steiner symmetrizations. It provides us with valuable insights for studying the relevant applications and conclusions of spherical Steiner symmetrizations. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
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13 pages, 286 KiB  
Article
Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions
by Meng Qin, Zhuohua Zhang, Rui Luo, Mengjie Ren and Denghui Wu
Axioms 2024, 13(5), 320; https://doi.org/10.3390/axioms13050320 - 11 May 2024
Viewed by 747
Abstract
In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb [...] Read more.
In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
31 pages, 389 KiB  
Article
The Vertex Gutman Index and Gutman Index of the Union of Two Cycles
by Yinzhen Mei and Hongli Miao
Axioms 2024, 13(4), 269; https://doi.org/10.3390/axioms13040269 - 18 Apr 2024
Viewed by 839
Abstract
The Wiener index is one of the most classic and widely used indicators in topology. It reflects the average distance of any node pair in the graph. It not only makes the boundaries of given graphs clearer but also continuously generates topological indices [...] Read more.
The Wiener index is one of the most classic and widely used indicators in topology. It reflects the average distance of any node pair in the graph. It not only makes the boundaries of given graphs clearer but also continuously generates topological indices that are more suitable for new fields, such as the Gutman index. The Wiener index and Gutman index are two important topological indices, which are commonly used to describe the characteristics of molecular structure. They are closely related to the physical and chemical properties of molecular compounds. And they are widely used to predict the physical and chemical properties and biological activity of molecular compounds. In this paper, we study the vertex Gutman index and Gutman index and describe the structural characteristics of all cases of two simple cycles intersecting. We comprehensively analyze the Gutman index and vertex Gutman index in these cases in detail by means of classification discussion and analogical reasoning and characterize their maximum and minimum accordingly. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
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16 pages, 318 KiB  
Article
An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies
by Xiangyang Han, Senlin Wu and Longzhen Zhang
Axioms 2024, 13(2), 132; https://doi.org/10.3390/axioms13020132 - 19 Feb 2024
Viewed by 1066
Abstract
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global [...] Read more.
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
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