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.

Starshapedness is a generalization of convexity. A set C is convex if ∀x ∈ C and ∀y ∈ C the segment [x : y] ⊂ C. On the other hand, a set S is starshaped if ∃y ∈ S such that ∀x ∈ S the segment [x : y] ⊂ S. Due to these closely related definitions, convex and starshaped sets have many similarities, but there are also some striking differences. In this paper we continue our studies of such similarities and differences. Our main goal is to get characterizations of starshapedness and, further on, to describe a starshaped set and its kernel by means of cones included in its complement. Classification (2000). 52A07, 52A20, 52A30. Mathematics SubjectWe shall work in Euclidean space R d , and we use standard concepts from set theory, topology, linear algebra, and convexity. Although we use the notation of our former paper [4], we repeat here definitions of some basic notions, for the sake of convenience. If A ⊂ R d is any set, its complement will be denoted by A´. If x is a point and X is a set, we shall write x ∪ X instead of the more clumsy {x} ∪ X. If a, b are different points, by [a : b], [a : b) and (a : b) we shall denote the segment with endpoints a and b, the half-line or ray with origin a through b, and the line through a and b, respectively. The replacement of [ by ] in [a : b] or [a : b) simply means that the endpoint or origin a does not belong to the segment or half-line; analogously for the replacement of ] by [ in [a : b].

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