Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

In summary, we discussed a Hilbert space H and a closed convex subset K of H, with x0 ∈ K. It was shown that Nk(x0) is the set of all y in K that satisfy the condition ⟨y,x-x0⟩ ≤ 0 for all x ∈ K. Additionally, we considered the specific case where K is the unit disc and x0 = (0,0). In this case, Nk((0,0)) is just the single point (0,0). Lastly, we looked at the scenario where x0 = (0,1), and it was determined that Nk((0,1)) consists of all y in K with the form (0,c) where c ≥
  • #1
moh salem
21
0
[tex]Let \text{ } H \text{ }be \text{ }a \text{ } Hilbert \text{ } space, \text{ }K \text{ }be \text{ }a \text{ }closed\text{ }convex\text{ }subset \text{ } of \text{ }H \text{ }and \text{ }x_{0}\in K. \text{ }Then \\N_{K}(x_{0}) =\{y\in K:\langle y,x-x_{0}\rangle \leq 0,\forall x\in K\} .\text{ }Hence, \text{ }if \text{ }K=\left\{ (x,y):x^{2}+y^{2}\leq 1\right\}\\ is \text{ }closed\text{ } and\text{ }convex, \text{ }find \text{ }N_{K}((0,0))? [/tex]
Thanks.
 
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  • #2
If x0 = (0,0), then you have a two dimensional Euclidean space. K is the unit disc. Nk((0,0)) has just one point (0,0).

If you meant something else, I suggest you rewrite it.
 
  • #3
[tex]\text{ }yes,\text{ } x_{0} = (0,0)[/tex]
 
  • #4
Yes, I mean [tex]\text{ } x_{0} = (0,0).[/tex]
but, if [tex]\text{ } x_{0} = (0,1).[/tex] What is equal to [tex]N_{K}((0,1))?[/tex]
 
  • #5
I haven't checked it throughly, but it looks like all y in K of the form (0,c) where c ≥ 0.
 
  • #6
Thank u Mr. mathman.
 

Related to Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

1. What is the normal cone of a closed convex subset?

The normal cone of a closed convex subset in a Hilbert space is the set of all vectors that are orthogonal to the tangent cone of the subset at a given point. In other words, it is the set of all directions in which the subset does not have any "curvature".

2. Why is finding the normal cone important?

Finding the normal cone is important in optimization and mathematical analysis. It helps determine the direction of steepest descent for a convex function, which is useful in finding the minimum or maximum of the function.

3. How is the normal cone of a closed convex subset calculated?

The normal cone of a closed convex subset can be calculated using the separating hyperplane theorem. This theorem states that for any two non-intersecting convex sets, there exists a hyperplane that separates them. The normal cone is then the set of all vectors that are orthogonal to this separating hyperplane.

4. What properties does the normal cone have?

The normal cone of a closed convex subset has several important properties. First, it is always a closed convex cone. Second, it is always a subset of the dual space of the Hilbert space. Third, it is always a closed subset of the dual space. Finally, it is always non-empty, even if the subset itself is empty.

5. Can the normal cone of a closed convex subset be used in non-Hilbert spaces?

Yes, the concept of the normal cone can be extended to non-Hilbert spaces. In Banach spaces, the normal cone is defined as the set of all bounded linear functionals that are orthogonal to the tangent cone of the subset at a given point. However, the separating hyperplane theorem may not hold in all non-Hilbert spaces, so the calculation of the normal cone may be more complicated.

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