Convex Cone - Examples of Convex Cones

Examples of Convex Cones

Given a closed, convex subset K of V, the normal cone to the set K at the point x in K is given by

$N_K(x) = \{ p \in V | \langle p, x - x^* \rangle \geq 0, \forall x^* \in K \}.$

Given a closed, convex subset K of V, the tangent cone (or contingent cone) to the set K at the point x is given by

$T_K(x) = \overline{\bigcup_{h>0} \frac{1}{h} (K-x)}.$

Both the normal and tangent cone have the property of being closed and convex. They are important concepts in the fields of convex optimization, variational inequalities and projected dynamical systems.

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