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Brouwer Fixed Point Theorem

The Brouwer fixed point theorem states that any continuous function G:B^n->B^n has a fixed point, where

B^n={x in R^n:x_1^2+...+x_n^2<=1}

is the unit closed ball in R^n.

See also

Ball, Closed Ball, Fixed Point Theorem, Map Fixed Point

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References

Harvey Mudd College Mathematics Department. "Mudd Math Fun Facts: Brouwer Fixed Point Theorem." https://math.hmc.edu/funfacts/brouwer-fixed-point-theorem/.Kannai, Y. "An Elementary Proof of the No Retraction Theorem." Amer. Math. Monthly 88, 264-268, 1981.Milnor, J. W. Topology from the Differentiable Viewpoint. Charlottesville, VA: University Press of Virginia, p. 14, 1965.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub., p. 117, 1993.Numberphile. "The Fixed Point Theorem." Featuring C. Tucker and B. Sparks. Jun. 25, 2026. https://www.youtube.com/watch?v=6PmuWFWVKDE.Samelson, H. "On the Brouwer Fixed Point Theorem." Portugal. Math. 22, 189-191, 1963.

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Brouwer Fixed Point Theorem

Cite this as:

Weisstein, Eric W. "Brouwer Fixed Point Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrouwerFixedPointTheorem.html

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