In geometry, the tangent cone is a generalization of the notion of the tangent space (applied to manifolds in the usual differential geometry setting) to irregular surfaces, that may contain singularities such as angular corners.
Definitions in nonlinear analysis In nonlinear analysis, there are many definitions for a tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but they can differ on more general sets.
Clarke tangent cone Let A {\displaystyle A} be a nonempty closed subset of the Banach space X {\displaystyle X} . The Clarke's tangent cone to A {\displaystyle A} at x 0 ∈ A {\displaystyle x_{0}\in A} , denoted by T ^ A ( x 0 ) {\displaystyle {\widehat {T}}_{A}(x_{0})} consists of all vectors v ∈ X {\displaystyle v\in X} , such that for any sequence { t n } n ≥ 1 ⊂ R {\displaystyle \{t_{n}\}_{n\geq 1}\subset \mathbb {R} } tending to zero, and any sequence { x n } n ≥ 1 ⊂ A {\displaystyle \{x_{n}\}_{n\geq 1}\subset A} tending to x 0 {\displaystyle x_{0}} , there exists a sequence { v n } n ≥ 1 ⊂ X {\displaystyle \{v_{n}\}_{n\geq 1}\subset X} tending to v {\displaystyle v} , such that for all n ≥ 1 {\displaystyle n\geq 1} holds x n + t n v n ∈ A {\displaystyle x_{n}+t_{n}v_{n}\in A}
Clarke's tangent cone is always subset of the corresponding contingent cone (and coincides with it, when the set in question is convex). It has the important property of being a closed convex cone.
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