Tangent cone

The tangential or normal cone of a subset of a Euclidean space is in the geometry of a generalization of the notion of the tangent respectively of the normal vector of a set and thus enables the use of algebraic methods to non - differentiable geometric objects. Both the tangential and the normal taper cones in the sense of linear algebra, thus the description is warranted. Normal cone is also referred to as polar cone. The first unified version of the notion of tangent cone comes from the U.S. topologist Hassler Whitney in 1965, but this describes rather the edge of the cone in the modern sense. The modern definitions developed in the context of the theory of sets of positive reach and completed their program findings from differential geometry to a larger class of sets - as only differentiable manifolds - to transfer.

Definition

Let be a subset of a Euclidean space and a point that does not necessarily lie in itself, finally, denotes the Euclidean norm.

Then, ie, the amount

The tangent cone of and be polar cone

Is called at normal cone or polar cone.

Falls is located in the edge, the tangent cone is clear from all of the outgoing rays still make another point. The normal cone is the set of all vectors that include all of these rays with an angle of at least 90.degree.

Unit normal bundle

Based on these concepts can be - in analogy to Einheitstangentialbündel of differential geometry - define the unit normal bundle:

So it is the disjoint union of the outward normal vectors of length 1 to each point. This definition makes sense, since a cone is described completely by its respective unit vectors.

It should be noted that the unit normal bundle - in contrast to the tangent bundle - generally is not a vector bundle in the sense of vector analysis, since the normal cone are no sub- vector spaces in general.

Properties

• Both tangential and normal cones are closed cone.
• Furthermore, the normal cone is always convex.
• Between the cones the relation.
• Has positive range, so even applies. Especially must also be convex.
• In addition, it can be shown that is complete in this case.

Note: Some authors are therefore limited in the definition from the outset to points in the financial statements.

• Makes the margin of the tangent cone a subspace in - in this case, is necessary in the edge - so the point is differentiable and coincides with the classical tangent match. Even a hyperplane that is of codimension 1, it is generated by the corresponding normal vector.