Dual cone and polar cone

A set $C$ and its dual cone $C^{*}$.

A set $C$ and its polar cone $C^{o}$. The dual cone and the polar cone are symmetric to each other with respect to the origin.

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.

Dual cone

The dual cone $C^{*}$ of a subset $C$ in a linear space $X$, e.g. Euclidean space $\mathbb {R} ^{n}$, with topological dual space $X^{*}$ is the set

$C^{*}=\left\{y\in X^{*}:\langle y,x\rangle \geq 0\quad \forall x\in C\right\},$

where $\langle y,x\rangle$ is the duality pairing between $X$ and $X^{*}$, i.e. $\langle y,x\rangle =y(x)$.

$C^{*}$ is always a convex cone, even if $C$ is neither convex nor a cone.

Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as $\mathbb {R} ^{n}$ equipped with the Euclidean inner product) to be what is sometimes called the internal dual cone.

$C_{internal}^{*}:=\left\{y\in X:\langle y,x\rangle \geq 0\quad \forall x\in C\right\}.$

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: (i) $y$ is a normal at the origin of a hyperplane that supports $C$. (ii) $y$ and $C$ lie on the same side of that supporting hyperplane.
• $C^{*}$ is closed and convex.
• $C_{1}\subseteq C_{2}$ implies $C_{2}^{*}\subseteq C_{1}^{*}$.
• 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 $C$.

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 $\langle .,.\rangle$ 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 than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in $\mathbb {R} ^{n}$ with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base in $\mathbb {R} ^{n}$ is equal to its internal dual.

The nonnegative orthant of $\mathbb {R} ^{n}$ and the space of all positive semidefinite matrices 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 $\mathbb {R} ^{3}$ whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in $\mathbb {R} ^{3}$ 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

The polar of the closed convex cone $C$ is the closed convex cone $C^{o},$ and vice-versa.

For a set $C$ in $X$, the polar cone of $C$ is the set

$C^{o}=\left\{y\in X^{*}:\langle y,x\rangle \leq 0\quad \forall x\in C\right\}.$

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 for $C$.

See also

• Bipolar theorem
• Polar set

References

• Goh, C. J. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0-415-27479-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Boltyanski, V. G. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3-540-61341-2. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: publisher location (link)

• Ramm, A.G. (2000). Operator theory and its applications. Providence, R.I.: American Mathematical Society. ISBN 0-8218-1990-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)