In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
in the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an
such that for all
the
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
from
to
lies in
This definition is immediately generalizable to any
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
, or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
.
Intuitively, if one thinks of
as a region surrounded by a wall,
is a star domain if one can find a vantage point
in
from which any point
in
is within line-of-sight. A similar, but distinct, concept is that of a
radial set In mathematics, a subset A \subseteq X of a linear space X is radial at a given point a_0 \in A if for every x \in X there exists a real t_x > 0 such that for every t \in , t_x a_0 + t x \in A.
Geometrically, this means A is radial at a_0 if for e ...
.
Definition
Given two points
and
in a vector space
(such as
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
), the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of
is called the and it is denoted by
where
for every vector
A subset
of a vector space
is said to be
if for every
the closed interval
A set
is and is called a if there exists some point
such that
is star-shaped at
A set that is star-shaped at the origin is sometimes called a . Such sets are closed related to
Minkowski functionals.
Examples
* Any line or plane in
is a star domain.
* A line or a plane with a single point removed is not a star domain.
* If
is a set in
the set
obtained by connecting all points in
to the origin is a star domain.
* Any
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
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
A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a sa ...
-shaped figure is a star domain but is not convex.
* A
star-shaped polygon
In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible.
Formally, a polygon is star-shaped if there exists a poi ...
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
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
set, via a
straight-line homotopy. In particular, any star domain is a
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
set.
* Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio
the star domain can be dilated by a ratio
such that the dilated star domain is contained in the original star domain.
* The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
and
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of two star domains is not necessarily a star domain.
* A non-empty open star domain
in
is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to
* Given
the set
(where
ranges over all
unit length
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
scalars) is a
balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
whenever
is a star shaped at the origin (meaning that
and
for all
and
).
See also
*
*
*
*
*
*
*
*
*
*
References
* Ian Stewart, David Tall, ''Complex Analysis''. Cambridge University Press, 1983, ,
* C.R. Smith, ''A characterization of star-shaped sets'',
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an e ...
, Vol. 75, No. 4 (April 1968). p. 386, ,
*
*
*
External links
*
{{Convex analysis and variational analysis
Convex analysis
Euclidean geometry
Functional analysis
Linear algebra