In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set[1] or radially convex set) if there exists an such that for all the line segment from to lies in This definition is immediately generalizable to any real, or complex, vector space.

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.

Intuitively, if one thinks of as a region surrounded by a wall, is a star domain if one can find a vantage point in from which any point in is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Definition

edit

Given two points   and   in a vector space   (such as Euclidean space  ), the convex hull of   is called the closed interval with endpoints   and   and it is denoted by   where   for every vector  

A subset   of a vector space   is said to be star-shaped at   if for every   the closed interval   A set   is star shaped and is called a star domain if there exists some point   such that   is star-shaped at  

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

edit
  • Any line or plane in   is a star domain.
  • A line or a plane with a single point removed is not a star domain.
  • If   is a set in   the set   obtained by connecting all points in   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

edit
  • 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   the star domain can be dilated by a ratio   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   the set   (where   ranges over all unit length scalars) is a balanced set whenever   is a star shaped at the origin (meaning that   and   for all   and  ).
  • Diffeomorphism: A non-empty open star domain   in   is diffeomorphic to  
  • Binary operators: If   and   are star domains, then so is the Cartesian product  , and the sum  .[1]
  • Linear transformations: If   is a star domain, then so is every linear transformation of  .[1]

See also

edit

References

edit
  1. ^ a b c 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.
edit