About convex hull and fixed point

In summary, a convex hull is the outermost boundary of a set of points that forms a convex shape. It is calculated using a mathematical algorithm and has various applications in fields such as computer graphics and pattern recognition. A fixed point is a point on the convex hull that remains unchanged after applying a transformation or operation, and it must always lie on the convex hull.
  • #1
sapporozoe
12
0
First, I wonder whether I can put the post here...

Given
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...

What if we have a(x)={y in X:||y-x||>=1/2}? (My answer is x=1/2 or y=1/2)

Thanks.
 
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  • #2
What is the definition of convex hull? Of fixed point?
 
  • #3


The set of fixed points for a(x)={y in X:||y-x||>=1/4} is correct. This can be seen by visualizing the set a(x) on a graph. It forms a square with side length 1/2 centered at (1/2,1/2). The fixed points are the points on the boundary of this square, which are x=1/4, 3/4 and y=1/4, 3/4.

For the case of a(x)={y in X:||y-x||>=1/2}, the fixed points are x=1/2 and y=1/2. This can also be visualized on a graph, where the set a(x) forms a larger square with side length 1 centered at (1/2,1/2). The fixed points are the points on the boundary of this square, which are x=1/2 and y=1/2.

In general, for any value of the radius r in a(x)={y in X:||y-x||>=r}, the fixed points will always be on the boundary of the square formed by a(x), with x=r and y=r. This is because these points are the furthest possible points from the center (1/2,1/2) while still being contained within the square.

I hope this helps clarify the concept of fixed points and how they relate to the convex hull. Remember, the convex hull is the smallest convex set that contains all the points in a given set, and the fixed points are the points that remain unchanged when applying transformations to the set.
 

1. What is a convex hull?

A convex hull is the smallest convex shape that contains a set of points in a given space. In simpler terms, it is the outermost boundary that encloses a set of points, and it is always a convex shape, meaning that there are no "dents" or concave parts in the boundary.

2. How is the convex hull calculated?

The convex hull is calculated by using a mathematical algorithm that identifies the points on the outer boundary of the set and connects them to form a convex shape. There are different algorithms that can be used depending on the type of input data and the desired accuracy of the convex hull.

3. What are the applications of convex hulls?

Convex hulls have various applications in fields such as computer graphics, computer vision, and pattern recognition. They are commonly used for shape analysis, collision detection, and finding the shortest path between two points.

4. What is a fixed point in relation to convex hulls?

A fixed point is a point on the convex hull that remains unchanged after applying a transformation or operation to the set of points. It is useful for determining the stability of a shape or for finding a reference point for further calculations.

5. Can a fixed point be located outside of the convex hull?

No, a fixed point must always lie on the convex hull. This is because the convex hull is the smallest possible shape that contains all the points, so any point outside of the convex hull would not be part of the original set of points and therefore cannot be a fixed point.

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