Starshaped sets

被引：27

作者：

Hansen, G. ^{[1
]}

Herburt, I ^{[2
]}

Martini, H. ^{[3
]}

Moszynska, M. ^{[4
]}

机构：

[1] Univ Lujan, RA-6700 Lujan, Buenos Aires, Argentina

[2] Warsaw Univ Technol, Fac Math & Informat Sci, PL-00662 Warsaw, Poland

[3] TU Chemnitz, Fak Math, D-09107 Chemnitz, Germany

[4] Warsaw Univ, Inst Math, Ul Banacha 2, PL-02097 Warsaw, Poland

来源：

AEQUATIONES MATHEMATICAE | 2020年 / 94卷 / 06期

关键词：

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; L-p 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; KRASNOSELSKII-TYPE THEOREM; BUSEMANN-PETTY PROBLEM; FIXED-POINT THEOREMS; HELLY-TYPE THEOREM; INVERSE CURVATURE FLOWS; ALONG-RAYS PROPERTY; CROSS-SECTION BODY; TH MEAN-CURVATURE; INTERSECTION BODIES; CONVEX-BODIES;

D O I：

10.1007/s00010-020-00720-7

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：1001 / 1092

页数：92

共 560 条

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Abbas M, 2009, CARPATHIAN J MATH, V25, P141

[2]

AEPPLI A, 1960, P AM MATH SOC, V11, P826, DOI DOI 10.1090/S0002-9939-1960-0121567-1

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