Starshapedness vs. Convexity

Abstract

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].

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…Starshaped sets with the stronger condition at the boundary considered in Definition 3.1 are also studied in different contexts. See [28, p. 1008] and the references therein where these sets are also referred to as strongly star-shaped or radiative at p. See also [29] for a comparison of different properties concerning convex and star-shaped sets.…”

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…Star-shaped sets with the stronger condition at the boundary considered in Definition 3.1 are also studied in different contexts. See [28, p. 1008] and the references therein where these sets are also referred to as strongly star shaped or radiative at p. See also [29] for a comparison of different properties concerning convex and star-shaped sets.…”

Specifying the staircase kernel of a two-fold connected orthogonal polygon

“…A set C in the Euclidean space R n is convex if for any x ∈ C and any y ∈ C the line segment [x, y] joining x and y is contained in C. More generally, C is called starshaped if it is only required that there exists an x ∈ C such that [x, y] ⊂ C, for any y ∈ C. The union of all such points x is then called the kernel of the starshaped set C. Similarities and differences between these two concepts as well as many characterization theorems are presented in the expository paper (Hansen and Martini 2011). In particular, many properties of starshapedness are summarized there.…”