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^[1] or radially convex set) if there exists an $s_{0}\in S$ such that for all $s\in S,$ 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 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.

Definition

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

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

A set that is star-shaped at the origin is sometimes called a star set.^[2] Such sets are closely related to Minkowski functionals.

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.
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

Convexity: 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.
Closure and interior: The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
Contraction: Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
Shrinking: 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.^[3]
Union and intersection: The union or intersection of two star domains is not necessarily a star domain.
Balance: 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 $w\in W$ ).
Diffeomorphism: A non-empty open star domain $S$ in $\mathbb {R} ^{n}$ is diffeomorphic to $\mathbb {R} ^{n}.$
Binary operators: If $A$ and $B$ are star domains, then so is the Cartesian product $A\times B$ , and the sum $A+B$ .^[1]
Linear transformations: If $A$ is a star domain, then so is every linear transformation of $A$ .^[1]

References

^ ^a ^b ^c Braga de Freitas, Sinval; Orrillo, Jaime; Sosa, Wilfredo (2020-11-01). "From Arrow–Debreu condition to star shape preferences". Optimization. 69 (11): 2405–2419. doi:10.1080/02331934.2019.1576664. ISSN 0233-1934.
^ Schechter 1996, p. 303.
^ 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

Humphreys, Alexis. "Star convex". MathWorld.

[:0-1] Braga de Freitas, Sinval; Orrillo, Jaime; Sosa, Wilfredo (2020-11-01). "From Arrow–Debreu condition to star shape preferences". Optimization. 69 (11): 2405–2419. doi:10.1080/02331934.2019.1576664. ISSN 0233-1934.

[FOOTNOTESchechter1996303-2] Schechter 1996, p. 303.

[3] Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.

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