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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f \\colon S^n \\to \\R^n$ be a [[continuous map|0VGFHTM75Y]] on the [[$n$-sphere|T62369RFH6]] $S^n \\subset \\R^{n+1}$.\nThen there is a point $x \\in S^n$ with $f(x) = f(-x)$.\n\n## Remarks\n\nAn equivalent form says that there is no continuous odd map $g \\colon S^n \\to S^{n-1}$, that is, no map with $g(-x) = -g(x)$ for all $x$.\nIndeed, if $f(x) \\neq f(-x)$ for all $x$, then $g(x) = (f(x) - f(-x)) / \\norm{f(x) - f(-x)}$ is such a map.\nOne proof uses the fact that an odd map $S^{n-1} \\to S^{n-1}$ has odd [[degree|TMFG2P912R]].\nThe restriction of an odd map $S^n \\to S^{n-1}$ to the equator is an odd map that extends over a hemisphere.\nSo it is null-homotopic and has degree $0$, a contradiction.\n\nExample: for $n = 1$, let $h(x) = f(x) - f(-x)$ on the circle.\nThen $h(-x) = -h(x)$, so $h$ takes values of both signs or vanishes, and by the intermediate value theorem it has a zero.\n\nExample of use: at any moment there are two antipodal points on the Earth with the same temperature and the same air pressure, if both are continuous functions on $S^2$.\nThe theorem fails for maps $S^n \\to \\R^{n+1}$: the inclusion $S^n \\subset \\R^{n+1}$ takes different values at $x$ and $-x$.\n\n## Explanations\n\n### L1 human\n\nIf you squash a sphere flat into a space of one dimension less, two opposite points of the sphere always land on the same spot.\nFor a balloon pressed flat onto a table, some pair of opposite points ends up one on top of the other.\n\n### L2 human\n\nThe theorem is the starting point of the use of topology in combinatorics and geometry.\nIt implies the ham sandwich theorem, the Lusternik–Schnirelmann covering theorem, and Lovász's computation of the chromatic number of Kneser graphs.\nIts combinatorial counterpart is [[Tucker's lemma|PADZ7D9KBT]], from which it can also be proved.\n",
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