Brouwer Fixed Point Theorem

Any continuous Function $G:\Bbb{B}^n\to \Bbb{B}^n$ has a Fixed Point, where

$$\Bbb{B}^n=\{{\bf x}\in\Bbb{R}^n : {x_1}^2+\ldots+{x_n}^2\leq 1\}$$

is the unit $n$-Ball.

See also Ball, Fixed Point Theorem

References

Milnor, J. W. Topology from the Differentiable Viewpoint. Princeton, NJ: Princeton University Press, p. 14, 1965.