Tangential cone and normal cone

The tangential or normal cone of a subset of Euclidean space is in the geometry of a generalization of the concept of tangent or of the normal vector allows an amount, and thereby the application of algebraic methods on non- differentiable geometric objects. Both the tangential and the normal cone are cones in the sense of linear algebra , which justifies the name. The normal cone is also known as the polar cone . The first uniform version of the term tangential cone comes from the US topologist Hassler Whitney in 1965, but this rather described the edge of the cone in today's sense. The modern definitions developed in the context of the theory of sets of positive range and supplemented its program in order to be able to transfer knowledge from differential geometry to a larger class of sets - than just differentiable manifolds .

definition

Let be a subset of a Euclidean space and a point that does not necessarily have to lie in itself , after all denote the Euclidean norm . ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ ${\displaystyle a\in \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle \|\cdot \|}$

Then the amount is called

${\displaystyle \operatorname {Tan} (A;a):=\left\{u\in \mathbb {R} ^{n}|\exists (a_{i})_{i\in \mathbb {N} }\subseteq A\setminus \{a\}\colon a_{i}\to a\land {\frac {a_{i}-a}{\|a_{i}-a\|}}\to {\frac {u}{\|u\|}}\right\}\cup \{{\underline {0}}\}}$

the tangential cone of an and its polar cone ${\displaystyle A}$ ${\displaystyle a}$

${\displaystyle \operatorname {Nor} (A;a):=\operatorname {pol} \ \operatorname {Tan} (A;a)=\{v\in \mathbb {R} ^{n}|\forall u\in \operatorname {Tan} (A;a)\colon v^{T}u\leq 0\}}$

is called normal cone or polar cone from on . ${\displaystyle A}$ ${\displaystyle a}$

If it is in the edge , then the tangential cone clearly consists of all rays emanating from it that hit another point. The normal cone is then the set of all vectors that enclose an angle of at least 90 ° with all these rays . ${\displaystyle a}$ ${\displaystyle \partial A}$ ${\displaystyle a}$ ${\displaystyle A}$

Normal unit bundle

Building on these terms,  the normal unit bundle can be defined - in analogy to the unit tangential bundle of differential geometry :

${\displaystyle \operatorname {nor} A:=\{(a;{\vec {o}})\in \mathbb {R} ^{2n}|a\in A\ \land \ {\vec {o}}\in \operatorname {Nor} (A;a)\ \land \ \|{\vec {o}}\|=1\}=\bigsqcup _{a\in A}\left(\operatorname {Nor} (A;a)\cap S^{n-1}\right)}$

So it is the disjoint union of the outer normal vectors of length 1 to every point of . This definition makes sense, because each cone is completely described by its unit vectors. ${\displaystyle A}$

It should be noted that the normal unit bundle - in contrast to the tangential bundle - generally does not represent a vector space bundle in the sense of vector analysis , since the normal cones are usually not sub-vector spaces .

properties

• Both tangential and normal cones are closed cones.
• Furthermore, the normal cone is always convex .
• The relationship applies between the cones . ${\displaystyle \operatorname {pol} \ \operatorname {Nor} (A;a)\supseteq \operatorname {Tan} (A;a)}$
• Has a positive reach, it even applies . ${\displaystyle A}$ ${\displaystyle \operatorname {pol} \ \operatorname {Nor} (A;a)=\operatorname {Tan} (A;a)}$
  • In particular, it must then also be convex. ${\displaystyle \operatorname {Tan} (A;a)}$
  • It can also be shown that in this case im is closed. ${\displaystyle \operatorname {nor} A}$ ${\displaystyle \mathbb {R} ^{2n}}$
• If there is an internal point , the two cones degenerate into and ${\displaystyle a\in A^{\circ }}$ ${\displaystyle \operatorname {Tan} (A;a)=\mathbb {R} ^{n}}$ ${\displaystyle \operatorname {Nor} (A;a)=\{{\underline {0}}\}}$
• Is the other way around of separated, then vice versa: and ${\displaystyle a\in \mathbb {R} ^{n}\setminus {\overline {A}}}$ ${\displaystyle A}$ ${\displaystyle \operatorname {Tan} (A;a)=\{{\underline {0}}\}}$ ${\displaystyle \operatorname {Nor} (A;a)=\mathbb {R} ^{n}}$
• In optimization (mathematics) , tangential cones are used to derive optimality criteria. Usually, however, the linearized tangential cone is used because it is easier to handle.

Note: Some authors therefore limit their definition to points in the conclusion from the start . ${\displaystyle a\in {\overline {A}}}$

• If the edge of the tangential cone forms a sub-vector space in  - in this case it is necessarily in the edge  - the point is differentiable and corresponds to the classical tangential space . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle a}$ ${\displaystyle \partial A}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle \partial \operatorname {Tan} (A;a)}$ ${\displaystyle T_{a}A}$
  • If there is even a hyperplane , that is of codimension 1, then the corresponding normal vector is generated. ${\displaystyle \partial \operatorname {Tan} (A;a)}$ ${\displaystyle \operatorname {Nor} (A;a)}$

Individual evidence

1. ^ Hassler Whitney: Local properties of analytic varieties ; in: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), 205–244, Princeton University Press, Princeton, NJ, USA, 1965
2. ↑ Christoph Thäle: 50 Years sets of positive reach - A survey ; in: Surveys in Mathematics and its Applications Vol. 3, 123-165, 2008; Quoted from: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SMA/v03/v03.html Accessed July 1, 2012
3. ^ R. Tyrrell Rockafellar: Clarke's tangent cones and the boundaries of closed sets in ${\displaystyle \mathbb {R} ^{n}}$ ; in: Nonlinear Analysis, Theory, Methods, and Applications Vol. 3, 145-154, 1979; Quoted from: http://www.math.washington.edu/~rtr/papers/ Accessed July 1, 2012