Homework 11, Real Analysis

Due November 9

The fixed points of a continuous

$f:\mathbb B^n\to\mathbb B^n$ might not be interior.

The Brouwer fixed point theorem is false for the open ball.

Let

$K\subset\R^n$ be compact and convex, and

$f:K\to K$ continuous. Then f has a fixed point.

Let

$K\subset\R^n$ be compact and convex with

$C^1$ boundary,

$x_0\in K$ , and

$b:K\setminus\{x_0\}\to\partial K$ given by the intersection point of the line from

$x_0$ to

$x$ , on the side of

$x$ . Then

$b\in C^1$ .

Note that, if

$x_0$ is an interior point of

$K$ , then

$b$ is a retraction from

$K\setminus\{x_0\}$ onto

$\partial K$ .