In mathematics, a set \(S\) in the Euclidean space \(\mathbb {R} ^{n}\) is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an \({\displaystyle s_{0}\in S}\) such that for all \({\displaystyle s\in S,}\) 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 of 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.
Examples
- Any line or plane in \(\mathbb {R} ^{n}\) 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 \({\displaystyle \mathbb {R} ^{n},}\) the set \({\displaystyle B=\{ta:a\in A,t\in [0,1]\}}\) 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 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
- 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 \({\displaystyle r<1,}\) the star domain can be dilated by a ratio \(r\) such that the dilated star domain is contained in the original star domain.
- The union and intersection of two star domains is not necessarily a star domain.
- A non-empty open star domain \(S\) in \(\mathbb {R} ^{n}\) is diffeomorphic to \(\mathbb{R} ^{n}.\)
- Given \({\displaystyle W\subseteq X,}\) the set \({\displaystyle \bigcap _{|u|=1}uW}\) (where \(u\) ranges over all unit length scalars) is a balanced set whenever \(W\) is a star shaped at the origin (meaning that \({\displaystyle 0\in W}\) and \({\displaystyle rw\in W}\) for all \(0 \leq r \leq 1\) and \(w\in W\)).
See also
- Absolutely convex set
- Absorbing set – Set that can be "inflated" to reach any point
- Art gallery problem
- Balanced set – Construct in functional analysis
- Bounded set (topological vector space) – Generalization of boundedness
- Convex set – In geometry, set that intersects every line into a single line segment
- Star polygon
- Symmetric set
References
- 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. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
External links
- Humphreys, Alexis. "Star convex". MathWorld.
