Homework 11, Real Analysis

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.


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