Star domain


In geometry, a set in the Euclidean space is called a star domain 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.
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

Given two points and in a vector space , the convex hull of is called the and it is denoted by
where for every vector
A subset of a vector space is said to be if for every the closed interval
A set is and is called a if there exists some point such that is star-shaped at
A set that is star-shaped at the origin is sometimes called a. Such sets are closely related to Minkowski functionals.

Examples

  • 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

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.Union and intersection: The union or intersection of two star domains is not necessarily a star domain.Balance: Given the set is a balanced set whenever is a star shaped at the origin.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.Linear transformations: If is a star domain, then so is every linear transformation of.