Tangent cone

In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. 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. Let be a nonempty closed subset of the Banach space . The Clarke's tangent cone to at , denoted by consists of all vectors , such that for any sequence tending to zero, and any sequence tending to , there exists a sequence tending to , such that for all holds 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. Let K be a closed convex subset of a real vector space V and ∂K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. It is a convex cone in V and can also be defined as the intersection of the closed half-spaces of V containing K and bounded by the supporting hyperplanes of K at x. The boundary TK of the solid tangent cone is the tangent cone to K and ∂K at x. If this is an affine subspace of V then the point x is called a smooth point of ∂K and ∂K is said to be differentiable at x and TK is the ordinary tangent space to ∂K at x. Let X be an affine algebraic variety embedded into the affine space , with defining ideal . For any polynomial f, let be the homogeneous component of f of the lowest degree, the initial term of f, and let be the homogeneous ideal which is formed by the initial terms for all , the initial ideal of I. The tangent cone to X at the origin is the Zariski closed subset of defined by the ideal . By shifting the coordinate system, this definition extends to an arbitrary point of in place of the origin.