Dual cone and polar cone
Dual cone and polar cone are paired ideas in convex analysis.
- Dual cone: In a real vector space X with an inner product ⟨•,•⟩, the dual cone of a subset C ⊆ X is
C* = { y ∈ X : ⟨y, x⟩ ≥ 0 for all x ∈ C }.
If you identify X with its dual via the inner product, you can think of C* as the set of directions that make a nonnegative inner product with every x in C. The dual cone is always a convex cone.
In a more general setting (topological vector spaces), the dual cone can be defined using continuous linear functionals:
C* = { φ ∈ X* : φ(x) ≥ 0 for all x ∈ C }.
- Polar cone: The polar cone of C is
C° = { φ ∈ X* : φ(x) ≤ 0 for all x ∈ C }.
This is exactly the negative of the dual cone: C° = −C*.
- Self-duality: A cone C is self-dual if, with some inner product on X, it equals its internal dual C*. Some cones remain self-dual after choosing the right inner product; others do not. Examples that are self-dual under an appropriate inner product include the nonnegative orthant in R^n, the cone of positive semidefinite matrices, and many ellipsoidal (“spherical”) cones. In R^3, cones based on a regular polygon with an odd number of vertices can also be self-dual under the right inner product.
- Quick intuition and examples:
- In R^n with the standard inner product, C* consists of all vectors that have a nonnegative dot product with every vector in C.
- The polar cone C° collects functionals that never produce a positive value on C.
- The two notions are related by C° = −C*.
- Common self-dual cones include the nonnegative orthant and the cone of positive semidefinite matrices.
- Notes: If C is a closed convex cone, its polar cone is well-behaved and aligns with the polar set concept often used in convex analysis.
This page was last edited on 3 February 2026, at 11:58 (CET).