Topological Fixed Point Exercises

by Scott Garrabrant, SamEisenstat2 min read17th Nov 201847 comments

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Exercises / Problem-SetsLogic & Mathematics Fixed Point Theorems
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Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

This is one of three sets of fixed point exercises. The first post in this sequence is here, giving context.

1. (1-D Sperner's lemma) Consider a path built out of  edges as shown. Color each vertex blue or green such that the leftmost vertex is blue and the rightmost vertex is green. Show that an odd number of the edges will be bichromatic.

2. (Intermediate value theorem) The Bolzano-Weierstrass theorem states that any bounded sequence in  has a convergent subsequence. The intermediate value theorem states that if you have a continuous function  such that  and , then there exists an  such that . Prove the intermediate value theorem. It may be helpful later on if your proof uses 1-D Sperner's lemma and the Bolzano-Weierstrass theorem

3. (1-D Brouwer fixed point theorem) Show that any continuous function  has a fixed point (i.e. a point  with ). Why is this not true for the open interval ?

4. (2-D Sperner's lemma) Consider a triangle built out of  smaller triangles as shown. Color each vertex red, blue, or green, such that none of the vertices on the large bottom edge are red, none of the vertices on the large left edge are green, and none of the vertices on the large right edge are blue. Show that an odd number of the small triangles will be trichromatic.

5. Color the all the points in the disk as shown. Let  be a continuous function from a closed triangle to the disk, such that the bottom edge is sent to non-red points, the left edge is sent to non-green points, and the right edge is sent to non-blue points. Show that  sends some point in the triangle to the center.

6. Show that any continuous function  from closed triangle to itself has a fixed point.

7. (2-D Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of  to itself has a fixed point. (You may use the fact that given any closed convex subset  of , the function from  to  which projects each point to the nearest point in  is well defined and continuous.)

8. Reflect on how non-constructive all of the above fixed-point findings are. Find a parameterized class of functions where for each , and the function  is continuous, but there is no continuous way to pick out a single fixed point from each function (i.e. no continuous function  such that  is a fixed point of  for all ).

9. (Sperner's lemma) Generalize exercises 1 and 4 to an arbitrary dimension simplex.

10. (Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of  to itself has a fixed point.

11. Given two nonempty compact subsets , the Hausdorff distance between them is the supremum 

 over all points in either subset of the distance from that point to the other subset. We call a set valued function  a continuous Hausdorff limit if there is a sequence  of continuous functions from  to  whose graphs, , converge to the graph of , in Hausdorff distance. Show that every continuous Hausdorff limit  from a compact convex subset of  to itself has a fixed point (a point  such that ).

12. Let  and  be nonempty compact convex subsets of . We say that a set valued function,  is a Kakutani function if the graph of , is closed, and  is convex and nonempty for all . For example, we could take  and  to be the interval , and we could have  send each  to , map  to the whole interval , and map  to . Show that every Kakutani function is a continuous Hausdorff limit. (Hint: Start with the case where  is a unit cube. Construct  by breaking  into small cubes of side length . Constuct a smaller cube of side length  within each  cube. Send each small  to the convex hull of the images of all points in the  cube with a continuous function, and glue these together with straight lines. Make sure you don't accidentally get extra limit points.)

13. (Kakutani fixed point theorem) Show that every Kakutani function from a compact convex subset of  to itself has a fixed point.


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