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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $D^n = \\set{x \\in \\R^n : \\norm{x} \\le 1}$ be the [[closed unit ball|0Q0H13G8ZB]].\nEvery [[continuous map|0VGFHTM75Y]] $f \\colon D^n \\to D^n$ has a fixed point: there is $x \\in D^n$ with $f(x) = x$.\n\n## Remarks\n\nThe idea of the proof: suppose $f(x) \\neq x$ for all $x$.\nSend $x$ to the point where the ray from $f(x)$ through $x$ leaves the ball; this is a continuous [[retraction|68PFSGP39B]] of $D^n$ onto the sphere $S^{n-1}$ that fixes $S^{n-1}$.\nNo such retraction exists by the [[no-retraction theorem|Z6JR1GN2RE]], because $H_{n-1}(S^{n-1}) \\neq 0$ for $n \\ge 2$ while $D^n$ is contractible; for $n = 1$ the [[intermediate value theorem|00KAFBR32E]] applied to $f(x) - x$ suffices.\nThe theorem is also the case of a closed ball in the [[Lefschetz fixed-point theorem|G44NQXCWT9]], since $L(f) = 1$ for every self-map of a contractible compact polyhedron.\nIn the language of the [[fixed-point property|JQDHHFXBF3]], it says that $D^n$ has this property.\nSo does every space homeomorphic to $D^n$, such as every compact [[convex set|02AS45T3H9]] of $\\R^n$ with nonempty interior.\nEach hypothesis matters: the translation $x \\mapsto x + 1$ of the non-compact space $\\R$ has no fixed point.\nThe rotation $z \\mapsto -z$ of the circle $S^1$, which is compact but has a hole, has none either.\n\n## Explanations\n\n### L1 human\n\nIf you continuously deform a disk and put it back on top of itself, at least one point ends up exactly where it started.\nStirring a cup of coffee gently, without breaking the surface, always leaves some point of the surface in its original place.\nThe result says nothing about which point it is; it only guarantees that one exists.\n\n### L2 human\n\nThe theorem is the basic existence tool behind many other fixed-point results.\nSchauder's theorem extends it to compact convex sets in normed spaces, and Kakutani's theorem to set-valued maps, which is how Nash proved the existence of equilibria in games.\nIts proofs connect several subjects: degree theory, homology, the no-retraction theorem, and combinatorial arguments through Sperner's lemma.\n",
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