![Polar cone illustration1](https://upload.wikimedia.org/wikipedia/commons/a/aa/Polar_cone_illustration1.svg)
Dual cone and polar cone are closely related concepts in
convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of som ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
.
Dual cone
In a vector space
The dual cone ''C'' of a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''C'' in a
linear 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 ...
''X'' over the
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) ...
s, e.g.
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 ...
R
''n'', with
dual space ''X'' is the set
:
where
is the
duality pairing
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
between ''X'' and ''X'', i.e.
.
''C'' is always a
convex cone, even if ''C'' is neither
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
nor a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
.
In a topological vector space
If ''X'' is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
over the real or complex numbers, then the dual cone of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'':
:
,
which is the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
* Polar climate, the c ...
of the set -''C''.
No matter what ''C'' is,
will be a convex cone.
If ''C'' ⊆ then
.
In a Hilbert space (internal dual cone)
Alternatively, many authors define the dual cone in the context of a real
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(such as R
''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.
:
Using this latter definition for ''C'', we have that when ''C'' is a cone, the following properties hold:
* A non-zero vector ''y'' is in ''C'' if and only if both of the following conditions hold:
#''y'' is a
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
at the origin of a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
that
supports ''C''.
#''y'' and ''C'' lie on the same side of that supporting hyperplane.
*''C'' is
closed and convex.
*
implies
.
*If ''C'' has nonempty interior, then ''C'' is ''pointed'', i.e. ''C*'' contains no line in its entirety.
*If ''C'' is a cone and the closure of ''C'' is pointed, then ''C'' has nonempty interior.
*''C'' is the closure of the smallest convex cone containing ''C'' (a consequence of the
hyperplane separation theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least on ...
)
Self-dual cones
A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to ''C''.
Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
This is slightly different from the above definition, which permits a change of inner product.
For instance, the above definition makes a cone in R
''n'' with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in R
''n'' is equal to its internal dual.
The nonnegative
orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutua ...
of R
''n'' and the space of all
positive semidefinite matrices
In mathematics, a symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of ...
are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones").
So are all cones in R
3 whose base is the convex hull of a regular polygon with an odd number of vertices.
A less regular example is the cone in R
3 whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
Polar cone
![Polar cone illustration](https://upload.wikimedia.org/wikipedia/commons/a/a8/Polar_cone_illustration.svg)
For a set ''C'' in ''X'', the polar cone of ''C'' is the set
:
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. ''C
o'' = −''C''.
For a closed convex cone ''C'' in ''X'', the polar cone is equivalent to the
polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X lying in the dual space X^.
The bipolar of a subset is the polar of A^, but lies ...
for ''C''.
See also
*
Bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the Polar set, polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions f ...
*
Polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X lying in the dual space X^.
The bipolar of a subset is the polar of A^, but lies ...
References
Bibliography
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{{Ordered topological vector spaces
Convex analysis
Convex geometry
Linear programming