Brouwer fixed point theorem

Theorem Let $\textbf{B}=\{x\in\mathbb{R}^{n}:\left\|x\right\|\leq 1\}$ be the closed unit ball in $\mathbb{R}^{n}$. Any continuous function $f:\textbf{B}\to\textbf{B}$ has a fixed point.

Notes

Shape is not important

The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace B in the formulation with a square or a triangle.

Compactness counts (a)

The theorem is not true if we drop a point from the interior of B. For example, the map $f(\vec{x})=\frac{1}{2}\vec{x}$ has the single fixed point at $0$; dropping it from the domain yields a map with no fixed points (http://planetmath.org/FixedPoint).

Compactness counts (b)

The theorem is not true for an open disk. For instance, the map $f(\vec{x})=\frac{1}{2}\vec{x}+(\frac{1}{2},0,\ldots,0)$ has its single fixed point on the boundary of B.

 Title Brouwer fixed point theorem Canonical name BrouwerFixedPointTheorem Date of creation 2013-03-22 12:44:34 Last modified on 2013-03-22 12:44:34 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 7 Author mathcam (2727) Entry type Theorem Classification msc 55M20 Classification msc 54H25 Classification msc 47H10 Related topic FixedPoint Related topic SchauderFixedPointTheorem Related topic TychonoffFixedPointTheorem Related topic KKMlemma Related topic KKMLemma