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Homework 11, Real Analysis

Due November 9

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

Let $latex K\subset\R^n$ be compact and convex with $latex C^1$ boundary, $latex x_0\in K$, and $latex b:K\setminus\{x_0\}\to\partial K$ given by the intersection point of the line from $latex x_0$ to $latex x$, on the side of $latex x$. Then $latex b\in C^1$.

Note that, if $latex x_0$ is an interior point of $latex K$, then $latex b$ is a retraction from $latex K\setminus\{x_0\}$ onto $latex \partial K$.