Star domain

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

In geometry, a set ${\displaystyle S}$ in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle s_{0}}$ to ${\displaystyle s}$ lies in ${\displaystyle S.}$ This definition is immediately generalizable to any real, or complex, vector space.

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.

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

Definition

Given two points ${\displaystyle x}$  and ${\displaystyle y}$  in a vector space ${\displaystyle X}$  (such as Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ ), the convex hull of ${\displaystyle \{x,y\}}$  is called the closed interval with endpoints ${\displaystyle x}$  and ${\displaystyle y}$  and it is denoted by

$${\displaystyle \left[x,y\right]~:=~\left\{tx+(1-t)y:0\leq t\leq 1\right\}~=~x+(y-x)[0,1],}$$

 

where ${\displaystyle z[0,1]:=\{zt:0\leq t\leq 1\}}$  for every vector ${\displaystyle z.}$ 

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

A set that is star-shaped at the origin is sometimes called a star set.^[1] Such sets are closed related to Minkowski functionals.

Examples

• Any line or plane in ${\displaystyle \mathbb {R} ^{n}}$  is a star domain.
• A line or a plane with a single point removed is not a star domain.
• If ${\displaystyle A}$  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 ${\displaystyle A}$  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 ${\displaystyle r}$  such that the dilated star domain is contained in the original star domain.^[2]
• The union and intersection of two star domains is not necessarily a star domain.
• A non-empty open star domain ${\displaystyle S}$  in ${\displaystyle \mathbb {R} ^{n}}$  is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$ 
• Given ${\displaystyle W\subseteq X,}$  the set ${\displaystyle \bigcap _{|u|=1}uW}$  (where ${\displaystyle u}$  ranges over all unit length scalars) is a balanced set whenever ${\displaystyle W}$  is a star shaped at the origin (meaning that ${\displaystyle 0\in W}$  and ${\displaystyle rw\in W}$  for all ${\displaystyle 0\leq r\leq 1}$  and ${\displaystyle w\in W}$ ).

See also

• Absolutely convex set – convex and balanced setPages displaying wikidata descriptions as a fallback
• 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
• Radial set
• Star polygon – Regular non-convex polygon
• Symmetric set – Property of group subsets (mathematics)

References

1. Schechter 1996, p. 303.
2. Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.

• 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.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 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

 

Wikimedia Commons has media related to Star-shaped sets.

• Humphreys, Alexis. "Star convex". MathWorld.