In geometry, a set S {\displaystyle S} in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s 0 ∈ S {\displaystyle s_{0}\in S} such that for all s ∈ S , {\displaystyle s\in S,} the line segment from s 0 {\displaystyle s_{0}} to s {\displaystyle s} lies in S . {\displaystyle S.} This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S {\displaystyle S} as a region surrounded by a wall, S {\displaystyle S} is a star domain if one can find a vantage point s 0 {\displaystyle s_{0}} in S {\displaystyle S} from which any point s {\displaystyle s} in S {\displaystyle S} is within line-of-sight. A similar, but distinct, concept is that of a radial set.
Definition Given two points x {\displaystyle x} and y {\displaystyle y} in a vector space X {\displaystyle X} (such as Euclidean space R n {\displaystyle \mathbb {R} ^{n}} ), the convex hull of { x , y } {\displaystyle \{x,y\}} is called the closed interval with endpoints x {\displaystyle x} and y {\displaystyle y} and it is denoted by
[ x , y ] := { t y + ( 1 − t ) x : 0 ≤ t ≤ 1 } = x + ( y − x ) [ 0 , 1 ] , {\displaystyle \left[x,y\right]~:=~\left\{ty+(1-t)x:0\leq t\leq 1\right\}~=~x+(y-x)[0,1],}
where z [ 0 , 1 ] := { z t : 0 ≤ t ≤ 1 } {\displaystyle z[0,1]:=\{zt:0\leq t\leq 1\}} for every vector z . {\displaystyle z.}
A subset S {\displaystyle S} of a vector space X {\displaystyle X} is said to be star-shaped at s 0 ∈ S {\displaystyle s_{0}\in S} if for every s ∈ S , {\displaystyle s\in S,} the closed interval
[ s 0 , s ] ⊆ S . {\displaystyle \left[s_{0},s\right]\subseteq S.}
A set S {\displaystyle S} is star shaped and is called a star domain if there exists some point s 0 ∈ S {\displaystyle s_{0}\in S} such that S {\displaystyle S} is star-shaped at s 0 . {\displaystyle s_{0}.}
A set that is star-shaped at the origin is sometimes called a star set. Such sets are closely related to Minkowski functionals.
Examples Any line or plane in R n {\displaystyle \mathbb {R} ^{n}} is a star domain. A line or a plane with a single point removed is not a star domain. If A {\displaystyle A} is a set in R n , {\displaystyle \mathbb {R} ^{n},} the set B = { t a : a ∈ A , t ∈ [ 0 , 1 ] } {\displaystyle B=\{ta:a\in A,t\in [0,1]\}} obtained by connecting all points in A {\displaystyle A} to the origin is a star domain. A cross-shaped figure is a star domain but is not convex. A star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.
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