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.

Highlights

  • AEMWhile convex geometry has a long history, going back even to ancient times (e.g., Archimedes) and to later contributors like Kepler, Euler, Cauchy, and Steiner, the geometry of starshaped sets is a younger field, and no historical overview exists

  • Fejes Toth, is whether any convex set is the kernel of some non-convex starshaped set. de Bruijn and Post [417] answered this question for the planar case, and Klee [301] gave a general answer with Theorem 52

  • The notion of radial function of starshaped sets is widely discussed in Sects. 0.7, 0.8 and 0.9 of Gardner’s monograph [204]; various properties and related notions are given there in a comprehensive way, concerning the more general definition of starshapedness given in Sect. 0.7 of this monograph

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Summary

Introduction

Dimension of the kernel of a starshaped set 12. Unions, and intersections of starshaped sets 15.

Basic notation and definitions
Starshaped sets and visibility
Star generators: representations of the kernel
Krasnosel’skii-type theorems
Asymptotic structure of starshaped sets
Separation of starshaped sets
10. Extremal structure of starshaped sets
11. Dimension of the kernel of a starshaped set
12. Admissible kernels of starshaped sets
13. Radial functions of starshaped sets
15. Spaces of starshaped sets
16. Selectors for star bodies
18. Extensions and generalizations
19. Applications and further topics
19.1. Discrete and computational geometry
19.2. Inequalities
19.3. Starshapedness in differential geometry
19.4. Starshaped sets and PDE
19.5. Starshapedness in fixed point theory
19.6. Starshaped sets in approximation theory
19.7. Applications of starshapedness in optimization
19.8. Further topics
Full Text
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