Star domain

"Star-shaped" redirects here. For the Blur documentary, see Starshaped.

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.

In geometry, 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 $s_{0}\in S$ such that for all $s\in S,$ the line segment from s₀ 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 s₀ 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 $\mathbb {R} ^{n},$ the set $B=\{ta:a\in A,t\in [0,1]\}$ 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 $r<1,$ 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 $W\subseteq X,$ the set $\bigcap _{|u|=1}uW$ (where $u$ ranges over all unit length scalars) is a balanced set whenever $W$ is a star shaped at the origin (meaning that $0\in W$ and $rw\in W$ 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 – Mathematical 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 – Regular non-convex 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. Vol. 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.