Published May 20, 2020 | Version v1
Publication Open

Starshaped sets

Creators

  • 1. National University of Luján
  • 2. Warsaw University of Technology
  • 3. Chemnitz University of Technology
  • 4. University of Warsaw
  • 5. Institute of Mathematics

Description

Abstract This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines.

⚠️ This is an automatic machine translation with an accuracy of 90-95%

Translated Description (Arabic)

الملخص هذه ورقة تفسيرية حول المفهوم الرياضي الأساسي للنجومية، مع التأكيد على الخصائص الهندسية والتحليلية والتوافقية والطوبوغرافية للمجموعات النجمية وقابليتها للتطبيق على نطاق واسع في العديد من المجالات الرياضية. قرر المؤلفون التعامل مع الموضوع بطريقة واسعة للغاية لأنهم ليسوا على دراية بأي منشورات شبيهة بالمسح ذات صلة تتعامل مع هذه الفكرة الطبيعية. مفهوم التحدب النجمي قريب جدًا من مفهوم التحدب، وهو مطلوب في مجالات مثل التحدب الكلاسيكي، والتحليل المحدب، والتحليل الوظيفي، والهندسة المنفصلة والتوافقية والحسابية، والهندسة التفاضلية، ونظرية التقريب، و PDE، والتحسين ؛ يرتبط ارتباطًا وثيقًا بمفاهيم مثل الوظائف الشعاعية، ووظائف القسم، والرؤية، والمخاريط (الداعمة)، والحبات، والازدواجية، وغيرها الكثير. نقدم بطريقة مفصلة العديد من التعريفات والنظريات حول الخصائص الأساسية للمجموعات النجمية، تليها مناقشات شبيهة بالمسح للنتائج ذات الصلة. في نهاية المقال، نقوم بالإضافة إلى ذلك بمسح مجموعة واسعة من التطبيقات في بعض التخصصات المذكورة أعلاه.

Translated Description (French)

Résumé Il s'agit d'un document exposatif sur la notion mathématique fondamentale de la forme des étoiles, mettant l'accent sur les propriétés géométriques, analytiques, combinatoires et topologiques des ensembles en forme d'étoiles et leur large applicabilité dans de nombreux domaines mathématiques. Les auteurs ont décidé d'aborder le sujet de manière très large car ils ne sont au courant d'aucune publication similaire à une enquête traitant de cette notion naturelle. Le concept de forme d'étoile est très proche de celui de convexité, et il est nécessaire dans des domaines comme la convexité classique, l'analyse convexe, l'analyse fonctionnelle, la géométrie discrète, combinatoire et computationnelle, la géométrie différentielle, la théorie de l'approximation, la PDE et l'optimisation ; il est fortement lié à des notions comme les fonctions radiales, les fonctions de section, la visibilité, les cônes (de support), les noyaux, la dualité et bien d'autres. Nous présentons de manière détaillée de nombreuses définitions et théorèmes sur les propriétés de base des ensembles en forme d'étoiles, suivis de discussions de type enquête sur les résultats connexes. À la fin de l'article, nous examinons en outre un large éventail d'applications dans certaines des disciplines mentionnées ci-dessus.

Translated Description (Spanish)

Resumen Este es un documento expositivo sobre la noción matemática fundamental de la forma estelar, haciendo hincapié en las propiedades geométricas, analíticas, combinatorias y topológicas de los conjuntos con forma estelar y su amplia aplicabilidad en muchos campos matemáticos. Los autores decidieron abordar el tema de una manera muy amplia, ya que no conocen ninguna publicación similar a una encuesta relacionada que aborde esta noción natural. El concepto de forma de estrella está muy cerca del de convexidad, y es necesario en campos como la convexidad clásica, el análisis convexo, el análisis funcional, la geometría discreta, combinatoria y computacional, la geometría diferencial, la teoría de la aproximación, la PDE y la optimización; está fuertemente relacionado con nociones como funciones radiales, funciones de sección, visibilidad, conos (de soporte), núcleos, dualidad y muchos otros. Presentamos de manera detallada muchas definiciones y teoremas sobre las propiedades básicas de los conjuntos en forma de estrella, seguidos de discusiones similares a encuestas de resultados relacionados. Al final del artículo, además, examinamos un amplio espectro de aplicaciones en algunas de las disciplinas mencionadas anteriormente.

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Additional details

Additional titles

Translated title (Arabic)
أطقم على شكل نجوم
Translated title (French)
Ensembles starshaped
Translated title (Spanish)
Conjuntos Starshaped

Identifiers

Other
https://openalex.org/W4211082329
DOI
10.1007/s00010-020-00720-7

Is supplement to
https://openalex.org/W4378552409 (Other)
https://openalex.org/W4308588612 (Other)
https://openalex.org/W4303984019 (Other)
https://openalex.org/W2543137317 (Other)
https://openalex.org/W2423508375 (Other)
https://openalex.org/W2026568748 (Other)
https://openalex.org/W1793744123 (Other)
https://openalex.org/W1520256929 (Other)
https://openalex.org/W1031119431 (Other)
https://openalex.org/W1011247862 (Other)

GreSIS Basics Section

Is Global South Knowledge
Yes
Country
Argentina

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