Starshaped sets
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Aequationes Mathematicae
Starshaped sets ´ska G. Hansen, I. Herburt, H. Martini, and M. Moszyn
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. Mathematics Subject Classification. 51-02, 52-02, 52-99, 52A01, 52A07, 52A20, 52A21, 52A30, 52A35, 46B20, 47H10, 54E52, 54H25, 90C26, 90C48, 41A65, 35B30, 30C45, 32F17, 33C55, 43A90, 26A51, 26B25, 11H16. Keywords. Approximation theory, Asymptotic structure, Baire category, Banach spaces, Busemann–Petty problem, Centroid bodies, Cone, Cross-section body, Differential geometry, Discrete geometry, Dispensable point, Extremal structure, Extreme point, Fixed point theory, Fractal star body, Geometric inequalities, Geometry of numbers, Hausdorff metric, Helly’s theorem, Illumination, (Affine) inequalities, Infinity cone, Intersection bodies, Kakeya set, Kernel, Krasnosel’skii’s theorem, Krein–Milman theorem, Lp spaces, Minkowski’s theorem, Mirador, Optimization, Orlicz spaces, PDE, Radial function, Radial metric, Radial sum, Recession cone, Selectors, Separation, Spaces of starshaped sets, Star body, Star duality, Star generators, Star metric, Starlike sets, Starshaped hypersurfaces, Starshaped sets, Support cone, Valuations, Visibility.
G. Hansen et al.
AEM
1. Introduction While convex geometry has a long history (see, for instance, the bibliographies in [453] as well as in [185,232,234,292]), 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. The notion of starshapedness is a natural generalization of that of convexity. Its various versions appear later, starting with the beginning of the 20th century (see, e.g., [6,117]). An intensive development started in the 60’s, notably with the pioneering work of L´eopold Bragard (see [68–76]); many papers appeared also in the 70’s, 80’s, and later. This branch of geometry is still actively developing, and it has numerous applications.
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