Star domain


In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set 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.[1] Such sets are closed 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.
  • 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 any point in that set.
  • 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

  • The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
  • Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
  • 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.[2]
  • The union and intersection of two star domains is not necessarily a star domain.
  • A non-empty open star domain   in   is diffeomorphic to  
  • 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  ).

See also Edit

References Edit

  1. ^ Schechter 1996, p. 303.
  2. ^ Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.
  • Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ISBN 0-521-28763-4, MR0698076
  • C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR0227724, JSTOR 2313423
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.

External links Edit