more problems

[borisp]

Ok, so I am impressed at the problem solving ability of 2+2. So now I want to see if there are any problems I know that 2+2'ers can't solve in 5 minutes.

Some starters:

(A) Show that given a sphere and any five points on the surface of that sphere, there exists a closed hemisphere that contains four of them.

(B) Show that any continuous map f : B^n \to B^n has a fixed point; ie there exists p \in B^n such that f(p) = p. Here B^n denotes the Euclidean ball of radius B.

[thylacine]

[ QUOTE ]
Ok, so I am impressed at the problem solving ability of 2+2. So now I want to see if there are any problems I know that 2+2'ers can't solve in 5 minutes.

Some starters:

(A) Show that given a sphere and any five points on the surface of that sphere, there exists a closed hemisphere that contains four of them.

(B) Show that any continuous map f : B^n \to B^n has a fixed point; ie there exists p \in B^n such that f(p) = p. Here B^n denotes the Euclidean ball of radius B.

[/ QUOTE ]

(A) draw great circle through two points, then choose appropriate half.

(B) apply theorem, or repeat proof.

[borisp]

I guess some people who are familiar with theorems don't like to find their most elegant solutions. There are elementary arguments that verify Brouwer's fixed point theorem, although every one I know requires smoothness of f. I figured it might be interesting to pose it, in an effort to tease out one of those arguments from someone who hasn't had exposure to heavy machinery, and who wasn't aware that this is a standard theorem of algebraic topology.

And yes, the first problem is easy (from an old Putnam exam I think?) I use it to illustrate the concept of "proof" to my students on the first day of lecture in a proof based course.

[blah_blah]

the Brouwer theorem has a nice proof using PDE techniques which can be found in Evans' text /images/graemlins/smile.gif