Star domain

Short description: Property of point sets in Euclidean spaces

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 [math]\displaystyle{ S }[/math] in the Euclidean space [math]\displaystyle{ \R^n }[/math] is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an [math]\displaystyle{ s_0 \in S }[/math] such that for all [math]\displaystyle{ s \in S, }[/math] the line segment from [math]\displaystyle{ s_0 }[/math] to [math]\displaystyle{ s }[/math] lies in [math]\displaystyle{ S. }[/math] This definition is immediately generalizable to any real, or complex, vector space.

Intuitively, if one thinks of [math]\displaystyle{ S }[/math] as a region surrounded by a wall, [math]\displaystyle{ S }[/math] is a star domain if one can find a vantage point [math]\displaystyle{ s_0 }[/math] in [math]\displaystyle{ S }[/math] from which any point [math]\displaystyle{ s }[/math] in [math]\displaystyle{ S }[/math] is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Definition

Given two points [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] in a vector space [math]\displaystyle{ X }[/math] (such as Euclidean space [math]\displaystyle{ \R^n }[/math]), the convex hull of [math]\displaystyle{ \{x, y\} }[/math] is called the closed interval with endpoints [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] and it is denoted by [math]\displaystyle{ \left[x, y\right] ~:=~ \left\{t x + (1 - t) y : 0 \leq t \leq 1\right\} ~=~ x + (y - x) [0, 1], }[/math] where [math]\displaystyle{ z [0, 1] := \{z t : 0 \leq t \leq 1\} }[/math] for every vector [math]\displaystyle{ z. }[/math]

A subset [math]\displaystyle{ S }[/math] of a vector space [math]\displaystyle{ X }[/math] is said to be star-shaped at [math]\displaystyle{ s_0 \in S }[/math] if for every [math]\displaystyle{ s \in S, }[/math] the closed interval [math]\displaystyle{ \left[s_0, s\right] \subseteq S. }[/math] A set [math]\displaystyle{ S }[/math] is star shaped and is called a star domain if there exists some point [math]\displaystyle{ s_0 \in S }[/math] such that [math]\displaystyle{ S }[/math] is star-shaped at [math]\displaystyle{ s_0. }[/math]

A set that is star-shaped at the origin is sometimes called a star set. Such sets are closed related to Minkowski functionals.

Examples

• Any line or plane in [math]\displaystyle{ \R^n }[/math] is a star domain.
• A line or a plane with a single point removed is not a star domain.
• If [math]\displaystyle{ A }[/math] is a set in [math]\displaystyle{ \R^n, }[/math] the set [math]\displaystyle{ B = \{t a : a \in A, t \in [0, 1]\} }[/math] obtained by connecting all points in [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ r \lt 1, }[/math] the star domain can be dilated by a ratio [math]\displaystyle{ r }[/math] 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 [math]\displaystyle{ S }[/math] in [math]\displaystyle{ \R^n }[/math] is diffeomorphic to [math]\displaystyle{ \R^n. }[/math]
• Given [math]\displaystyle{ W \subseteq X, }[/math] the set [math]\displaystyle{ \bigcap_{|u|=1} u W }[/math] (where [math]\displaystyle{ u }[/math] ranges over all unit length scalars) is a balanced set whenever [math]\displaystyle{ W }[/math] is a star shaped at the origin (meaning that [math]\displaystyle{ 0 \in W }[/math] and [math]\displaystyle{ r w \in W }[/math] for all [math]\displaystyle{ 0 \leq r \leq 1 }[/math] and [math]\displaystyle{ w \in W }[/math]).

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 whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Radial set
• Star polygon – Regular non-convex polygon
• Symmetric set – Property of group subsets (mathematics)

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 (January 1, 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. https://archive.org/details/functionalanalys00rudi. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 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

• Humphreys, Alexis. "Star convex". http://mathworld.wolfram.com/StarConvex.html.