Generalize the proof of the Brouwer theorem to an arbitrary compact convex set as follows. Let f: S → S be a continuous operator on a nonempty, compact, convex subset of a finite-dimensional normed linear space.
1. Show that there exists a simplex T containing S.
2. By exercise 3.74, there exists a continuous retraction r: T → S. Show that f ∘ r: T → T and has a fixed point in x* ∊ T.
3. Show that x* ∊ S and therefore f (x*) = x*.
Consequently f has a fixed point x*.