Star domain explained

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

Intuitively, if one thinks of

S as a region surrounded by a wall, S is a star domain if one can find a vantage point s _{0} in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Definition

Given two points

x and y in a vector space X (such as Euclidean space \R ^{n} ), the convex hull of \{x, y\} is called the and it is denoted by$$\backslash left[x,\; y\backslash right]\; ~:=~\; \backslash left\backslash \; ~=~\; x\; +\; (y\; -\; x)\; [0,\; 1],$$where z [ 0 , 1 ] : = \{z t : 0 \leq t \leq 1 \} for every vector z.

A subset

S of a vector space X is said to be s _{0} \in S if for every s \in S, the closed interval\left[s _{0,} s\right] \subseteq S. A set S is and is called a if there exists some point s _{0} \in S such that S is star-shaped at s _{0.}

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

\R ^{n} is a star domain.

• A line or a plane with a single point removed is not a star domain.
• If

A is a set in \R ^{n,} the set B = \{t a : a \in A, t \in [ 0 , 1 ]\} obtained by connecting all points in A 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 each 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

• 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

r < 1 , the star domain can be dilated by a ratio r such that the dilated star domain is contained in the original star domain.^[1]

• The union and intersection of two star domains is not necessarily a star domain.
• A non-empty open star domain

S in \R ^{n} is diffeomorphic to \R ^{n.}

• Given

W \subseteq X, the set c ap _{|u|=1} u W (where u ranges over all unit length scalars) is a balanced set whenever W is a star shaped at the origin (meaning that 0 \in W and r w \in W for all 0 \leq r \leq 1 and w \in W ).

References

• Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983,,
• C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386,,

Notes and References

1. Web site: Drummond-Cole. Gabriel C.. What polygons can be shrinked into themselves?. Math Overflow. 2 October 2014.