Also, if we have a proof for #6 there’s a pleasant method for #7 that should work in any dimension:
We take our closed convex set S that has the bounded function h:S→S . We take a triangle T that covers S so that any point in S is also in T .
Now we define a new function h′:T→T such that h′(x)=h(cs(x)) where cs(x) is the function that maps x to the nearest point in S.
By #6 we know that h′ has a fixed point, since cs is continuous. We know that the fixed point of h′ cannot lie outside S because the range of h′ is S. This means h′ has a fixed point within S and since for x∈S, h(x)=h′(x), h has a fixed point.
I constructed a large triangle around the convex shape with the center somewhere in the interior. I then projected each point in the convex shape from the center towards the edge of the triangle in a proportional manner. ie. The center stays where it is, the points on the edge of the convex shape are projected to the edge of the triangle and a point 1/x of the distance from the center to the edge of the convex shape is 1/x of the distance from the center to the edge of the triangle.
Cleanest solution I can find for #8:
ft(x)=11+e10(t−x)
Also, if we have a proof for #6 there’s a pleasant method for #7 that should work in any dimension:
We take our closed convex set S that has the bounded function h:S→S . We take a triangle T that covers S so that any point in S is also in T .
Now we define a new function h′:T→T such that h′(x)=h(cs(x)) where cs(x) is the function that maps x to the nearest point in S.
By #6 we know that h′ has a fixed point, since cs is continuous. We know that the fixed point of h′ cannot lie outside S because the range of h′ is S. This means h′ has a fixed point within S and since for x∈S, h(x)=h′(x), h has a fixed point.
On my approach:
I constructed a large triangle around the convex shape with the center somewhere in the interior. I then projected each point in the convex shape from the center towards the edge of the triangle in a proportional manner. ie. The center stays where it is, the points on the edge of the convex shape are projected to the edge of the triangle and a point 1/x of the distance from the center to the edge of the convex shape is 1/x of the distance from the center to the edge of the triangle.