About convex hull and fixed point

[sapporozoe]

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.

 

[EnumaElish]

What is the definition of convex hull? Of fixed point?

 

[tacman]

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.