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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. High Quality Content by WIKIPEDIA articles! 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. Let K be a closed convex subset of a real vector space V and aeCK be the boundary of K. The solid tangent cone to K at a point x aee aeCK 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 aeCK at x. If this is an affine subspace of V then the point x is called a smooth point of aeCK and aeCK is said to be differentiable at x and TK is the ordinary tangent space to aeCK at x.

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