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## Due November 9

### Problem 1

The fixed points of a continuous $f:\mathbb B^n\to\mathbb B^n$ might not be interior.

### Problem 2

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

### Problem 3

Let $K\subset\R^n$ be compact and convex, and $f:K\to K$ continuous. Then f has a fixed point.

### Problem 4

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 $\partial K$.