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

The **dual cone** *C ^{*}* of a subset

where is the duality pairing between *X* and *X ^{*}*, i.e. .

*C ^{*}* is always a convex cone, even if

If *X* is a topological vector space 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*:

- ,
^{[1]}

which is the polar of the set -*C*.^{[1]}
No matter what *C* is, will be a convex cone.
If *C* ⊆ {0} then .

Alternatively, many authors define the dual cone in the context of a real Hilbert space (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

- A non-zero vector
*y*is in*C*if and only if both of the following conditions hold:^{*}

*y*is a normal at the origin of a hyperplane 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)

A cone *C* in a vector space *X* is said to be *self-dual* if *X* can be equipped with an inner product ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to *C*.^{[3]}
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 of **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 **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.

For a set *C* in *X*, the **polar cone** of *C* is the set^{[4]}

It can be seen that the polar cone is equal to the negative of the dual cone, i.e. *C ^{o}* = −

For a closed convex cone *C* in *X*, the polar cone is equivalent to the polar set for *C*.^{[5]}

- ^
^{a}^{b}Schaefer & Wolff 1999, pp. 215–222. **^**Boyd, Stephen P.; Vandenberghe, Lieven (2004).*Convex Optimization*(pdf). Cambridge University Press. pp. 51–53. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.**^**Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.**^**Rockafellar, R. Tyrrell (1997) [1970].*Convex Analysis*. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6.**^**Aliprantis, C.D.; Border, K.C. (2007).*Infinite Dimensional Analysis: A Hitchhiker's Guide*(3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

- Boltyanski, V. G.; Martini, H.; Soltan, P. (1997).
*Excursions into combinatorial geometry*. New York: Springer. ISBN 3-540-61341-2. - Goh, C. J.; Yang, X.Q. (2002).
*Duality in optimization and variational inequalities*. London; New York: Taylor & Francis. ISBN 0-415-27479-6. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Ramm, A.G. (2000). Shivakumar, P.N.; Strauss, A.V. (eds.).
*Operator theory and its applications*. Providence, R.I.: American Mathematical Society. ISBN 0-8218-1990-9. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

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