Dual cone

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In convex analysis, the dual cone C^* \ of a set C \subset H, where H \ is a Hilbert space, is the set

C^* = \left \{y\in H: y \cdot x \geq 0 \quad \forall x\in C \right \}

C^* \ is always a convex cone, even if C \ is neither convex nor a cone. The set C \ is usually a cone in \mathbb{R}^n, in which case y \in C^* if and only if -y \ is the normal of a hyperplane that supports C \ at the origin. When C \ is a cone, the following properties hold:

  • 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.
  • If the closure of C \ is pointed then C^* \ has nonempty interior.
  • C^{**} \ is the closure of the convex hull of C \.


A cone is said to be self-dual if C = C^* \. The nonnegative orthant of \mathbb{R}^n and the space of all positive semidefinite matrices are self-dual.

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