Star domain

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A star domain (equivalently, a star-convex or star-shaped set) is not necessarily convex in the ordinary sense.
An annulus is not a star domain.

In geometry, a set S {\displaystyle S} {\displaystyle S} in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set[1] or radially convex set) if there exists an s 0 ∈ S {\displaystyle s_{0}\in S} {\displaystyle s_{0}\in S} such that for all s ∈ S , {\displaystyle s\in S,} {\displaystyle s\in S,} the line segment from s 0 {\displaystyle s_{0}} {\displaystyle s_{0}} to s {\displaystyle s} {\displaystyle s} lies in S . {\displaystyle S.} {\displaystyle S.} This definition is immediately generalizable to any real, or complex, vector space.

Intuitively, if one thinks of S {\displaystyle S} {\displaystyle S} as a region surrounded by a wall, S {\displaystyle S} {\displaystyle S} is a star domain if one can find a vantage point s 0 {\displaystyle s_{0}} {\displaystyle s_{0}} in S {\displaystyle S} {\displaystyle S} from which any point s {\displaystyle s} {\displaystyle s} in S {\displaystyle 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} {\displaystyle x} and y {\displaystyle y} {\displaystyle y} in a vector space X {\displaystyle X} {\displaystyle X} (such as Euclidean space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}}), the convex hull of { x , y } {\displaystyle \{x,y\}} {\displaystyle \{x,y\}} is called the closed interval with endpoints x {\displaystyle x} {\displaystyle x} and y {\displaystyle 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],} {\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\}} {\displaystyle z[0,1]:=\{zt:0\leq t\leq 1\}} for every vector z . {\displaystyle z.} {\displaystyle z.}

A subset S {\displaystyle S} {\displaystyle S} of a vector space X {\displaystyle X} {\displaystyle X} is said to be star-shaped at s 0 ∈ S {\displaystyle s_{0}\in S} {\displaystyle s_{0}\in S} if for every s ∈ S , {\displaystyle s\in S,} {\displaystyle s\in S,} the closed interval [ s 0 , s ] ⊆ S . {\displaystyle \left[s_{0},s\right]\subseteq S.} {\displaystyle \left[s_{0},s\right]\subseteq S.} A set S {\displaystyle 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} {\displaystyle s_{0}\in S} such that S {\displaystyle S} {\displaystyle S} is star-shaped at s 0 . {\displaystyle s_{0}.} {\displaystyle s_{0}.}

A set that is star-shaped at the origin is sometimes called a star set.[2] Such sets are closely related to Minkowski functionals.

Examples

  • Any line or plane in R n {\displaystyle \mathbb {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} {\displaystyle A} is a set in R n , {\displaystyle \mathbb {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]\}} {\displaystyle B=\{ta:a\in A,t\in [0,1]\}} obtained by connecting all points in A {\displaystyle 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.

Properties

  • Convexity: any non-empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to each point in that set.
  • Closure and interior: The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
  • Contraction: Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
  • Shrinking: Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio r < 1 , {\displaystyle r<1,} {\displaystyle r<1,} the star domain can be dilated by a ratio r {\displaystyle r} {\displaystyle r} such that the dilated star domain is contained in the original star domain.[3]
  • Union and intersection: The union or intersection of two star domains is not necessarily a star domain.
  • Balance: Given W ⊆ X , {\displaystyle W\subseteq X,} {\displaystyle W\subseteq X,} the set ⋂ | u | = 1 u W {\displaystyle \bigcap _{|u|=1}uW} {\displaystyle \bigcap _{|u|=1}uW} (where u {\displaystyle u} {\displaystyle u} ranges over all unit length scalars) is a balanced set whenever W {\displaystyle W} {\displaystyle W} is a star shaped at the origin (meaning that 0 ∈ W {\displaystyle 0\in W} {\displaystyle 0\in W} and r w ∈ W {\displaystyle rw\in W} {\displaystyle rw\in W} for all 0 ≤ r ≤ 1 {\displaystyle 0\leq r\leq 1} {\displaystyle 0\leq r\leq 1} and w ∈ W {\displaystyle w\in W} {\displaystyle w\in W}).
  • Diffeomorphism: A non-empty open star domain S {\displaystyle S} {\displaystyle S} in R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} is diffeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.} {\displaystyle \mathbb {R} ^{n}.}
  • Binary operators: If A {\displaystyle A} {\displaystyle A} and B {\displaystyle B} {\displaystyle B} are star domains, then so is the Cartesian product A × B {\displaystyle A\times B} {\displaystyle A\times B}, and the sum A + B {\displaystyle A+B} {\displaystyle A+B}.[1]
  • Linear transformations: If A {\displaystyle A} {\displaystyle A} is a star domain, then so is every linear transformation of A {\displaystyle A} {\displaystyle A}.[1]

See also

References

  1. Braga de Freitas, Sinval; Orrillo, Jaime; Sosa, Wilfredo (2020-11-01). "From Arrow–Debreu condition to star shape preferences". Optimization. 69 (11): 2405–2419. doi:10.1080/02331934.2019.1576664. ISSN 0233-1934.
  2. Schechter 1996, p. 303.
  3. Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.