Star-Shaped Cross Order

In geometry, a set $S$ in the Euclidean space $\R^n$ 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 $s_0$ 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 $\R^n$), the convex hull of $\$ is called the and it is denoted by
\left , y\right~:=~ \left\ ~=~ x + (y - x) , 1
where $z$, 1:= \ for every vector $z.$

A subset $S$ of a vector space $X$ is said to be $s_0 \in S$ if for every $s \in S,$ the closed interval
\left _0, s\right\subseteq S.
A set $S$ is and is called a 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 . Such sets are closely related to Minkowski functionals.

Examples

* Any line or plane in $\R^n$ 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 $\R^n,$ the set $B = \$ 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.
* Union and intersection: The union or intersection of two star domains is not necessarily a star domain.
* Balance: Given $W \subseteq X,$ the set $\bigcap_ u W$ (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 $r w \in W$ for all $0 \leq r \leq 1$ and $w \in W$).
* Diffeomorphism: A non-empty open star domain $S$ in $\R^n$ is diffeomorphic to $\R^n.$
* Binary operators: If $A$ and $B$ are star domains, then so is the Cartesian product $A\times B$, and the sum $A + B$.
* Linear transformations: If $A$ is a star domain, then so is every linear transformation of $A$.

See also

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* Star-shaped preferences

References

* Ian Stewart, David Tall, ''Complex Analysis''. Cambridge University Press, 1983, ,
* C.R. Smith, ''A characterization of star-shaped sets'', American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, ,
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External links

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{{Convex analysis and variational analysis

Convex analysis
Euclidean geometry
Functional analysis
Linear algebra