About convex hull and fixed point

In summary, the set of fixed points for this problem is the set of points outside the circle centered at (x_0,y_0) with radius 1/4. The convex hull of a(x) is the smallest convex set containing this set, and by Kakutani's theorem, there exists at least one fixed point for this convex hull. However, the convex hull changes depending on the starting point (x_0,y_0), so the set of fixed points may vary. The concept of fixed points is important in this problem because it allows us to find a solution to the problem using Kakutani's theorem.
  • #1
sapporozoe
12
0
X=[0,1]^2

a(x)={y in X:||y-x||>=1/4}

b(x)is the convex hull of a(x).

Identify the set of fixed points.

My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure...

Thanks.
 
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  • #2
If you take x= [itex](x_0,y_0)[/itex], y= (x,y), then [itex]||x-y||\ge 1/4[/itex] becomes [itex]\sqrt{(x-x_0)^2+ (y-y_0)^2}\ge 1/4}[/itex] which is the same as [itex](x-x_0)^2+ (y-y_0)^2\ge 1/16[/itex], the set of points outside the circle centered at [itex](x_0,y_0)[/itex] with radius 1/4. The "convex hull" of a set, A, is the smallest convex set containing A. To be convex, for any two points the straight line segment between them must be in the set. Draw a picture and start "connecting points". It should be clear what the convex hull of this set is.

Now, my question is "what does this have to do with fixed points?" (maybe I missed that part of the course!). A fixed point, for a function f, is a point x such that f(x)= x. What function are you talking about?
 
  • #3
Thanks.
This case is a bit tricky. We are asked to find the fixed points under this correspondence...As you know, the original correspondence is not convex-valued and has no fixed points. Then the convex hull is convex-valued and by Kakutani's theorem there exists at least one fixed point.

I am not sure about my answer because the convex hull changes when we move from (0,0) (the convex hull is part of the disk) to (0,1/4) (the convex hull is the whole disk).
HallsofIvy said:
If you take x= [itex](x_0,y_0)[/itex], y= (x,y), then [itex]||x-y||\ge 1/4[/itex] becomes [itex]\sqrt{(x-x_0)^2+ (y-y_0)^2}\ge 1/4}[/itex] which is the same as [itex](x-x_0)^2+ (y-y_0)^2\ge 1/16[/itex], the set of points outside the circle centered at [itex](x_0,y_0)[/itex] with radius 1/4. The "convex hull" of a set, A, is the smallest convex set containing A. To be convex, for any two points the straight line segment between them must be in the set. Draw a picture and start "connecting points". It should be clear what the convex hull of this set is.

Now, my question is "what does this have to do with fixed points?" (maybe I missed that part of the course!). A fixed point, for a function f, is a point x such that f(x)= x. What function are you talking about?
 

What is a convex hull?

A convex hull is the smallest convex polygon that contains all given points in a set. In simpler terms, it is the shape formed by connecting the outermost points of a set of points.

What is the purpose of finding the convex hull?

The main purpose of finding the convex hull is to simplify complex shapes or point sets into a simpler and more manageable shape. It is often used in computer graphics, image processing, and data analysis.

How is the convex hull calculated?

The convex hull is typically calculated using algorithms such as Graham's scan, Jarvis march, or Quickhull. These algorithms use a divide and conquer approach to find the convex hull efficiently.

What is a fixed point?

A fixed point, also known as a stationary point, is a point in a function where the output or y-value remains the same even when the input or x-value is changed. In other words, the function has a fixed point when f(x) = x.

Why are fixed points important in mathematics?

Fixed points are important in mathematics because they help solve equations and prove the existence of solutions. They are also used in optimization problems, dynamical systems, and fractals.

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