Star region

star-shaped set with the center of its interior (green) is a star region ${\displaystyle x_{0},}$

In mathematics , a star-shaped set is understood to be a subset of the , to which there is a point (a star center or a star center ) from which all points of the set are "visible", that is, every straight connection from to any one Point is completely in . ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle x\in M}$ ${\displaystyle M}$

If a star-shaped set is also open , one speaks of a star region .

Formal definition

A set is called a star , if there is one , so that the route for all ${\displaystyle M\subseteq \mathbb {R} ^{n}}$ ${\displaystyle x_{0}\in M}$ ${\displaystyle x\in M}$

${\displaystyle [x_{0}\,x]=\left\{x_{0}+t(x-x_{0})\;\colon \;t\in [0,1]\right\}}$

is a subset of . ${\displaystyle M}$

Remarks

• Every non-empty convex set is star-shaped.
• The set of possible star centers is also called the center of the set. One can show that it is always convex. A set coincides with its center if and only if it is convex.
• Star-shaped sets are contractable . It follows:
• Star-shaped sets are simply connected , i.e. especially path-connected .
• A star region is an area .

See also

• Problem of the museum guards

literature

• Konrad Königsberger: Analysis 2. 1. Edition. Springer, 1993, ISBN 3-540-54723-1 , p. 345

Web links

• Star-shaped area in curve integrals and conservative vector fields on Mathematik Online (Uni Stuttgart)
• Eric W. Weisstein : star convex . In: MathWorld (English).
• star shaped on PlanetMath (English)