Star domain

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.

In mathematics, a set S in the Euclidean space Rⁿ is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an x₀ in S such that for all x in S the line segment from x₀ to x is in S. This definition is immediately generalizable to any real or complex vector space.

Intuitively, if one thinks of S as of a region surrounded by a wall, S is a star domain if one can find a vantage point x₀ in S from which any point x in S is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Examples[edit]

Any line or plane in Rⁿ 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 Rⁿ, 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.
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[edit]

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', i.e.: 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.^[1]
The union and intersection of two star domains is not necessarily a star domain.
A nonempty open star domain S in Rⁿ is diffeomorphic to Rⁿ.

References[edit]

^ 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

External links[edit]

Humphreys, Alexis. "Star convex". MathWorld.

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

[1]

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Star domain

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References[edit]

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