Star domain

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.

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

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 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 ${\displaystyle r<1,}$ 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 – 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 (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.

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