Star domain

"Star-shaped" redirects here. For the Blur documentary, see Starshaped.

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.

Examples

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

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

↑ 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, MR 0698076
C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR 0227724, JSTOR 2313423

External links

Wikimedia Commons has media related to Star-shaped sets.

Weisstein, Eric W. "Star convex". MathWorld.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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Star domain

Examples

Properties

See also

References

External links