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  "body": "\n## Statement\n\nLet $E$ be a real [[normed vector space|03J5RAKB30]] and let $K \\subseteq E$ be a nonempty [[convex set|02AS45T3H9]] that is compact in the [[weak topology|P6KHVSTPZV]] of $E$.\nLet $S$ be a [[semigroup|0XMEM406JM]], under composition, of maps $s \\colon K \\to K$ such that each $s \\in S$ is [[affine|0JF0TVYGWT]] and an [[isometry|02HPP5VCZH]] for the norm.\nThen the maps in $S$ have a common fixed point: there is $x \\in K$ with $s(x) = x$ for all $s \\in S$.\n\n## Remarks\n\nThe isometry condition can be weakened to the condition that $S$ is noncontracting: for $x \\ne y$ in $K$, the vector $0$ is not in the norm closure of $\\set{s(x) - s(y) : s \\in S}$.\nIn this form the theorem holds for locally convex spaces and weakly continuous affine maps.\nBy weak compactness of $K$, it suffices to treat finitely generated semigroups.\nFor these, one takes a fixed point of a suitable average of the generators.\nThe noncontracting condition and a geometric lemma on weakly compact convex sets show that it is fixed by each generator.\n\nThe isometry hypothesis cannot be dropped.\nOn $K = [0, 1]$, the constant maps $s_0(x) = 0$ and $s_1(x) = 1$ are affine and generate the semigroup $\\set{s_0, s_1}$, but they have no common fixed point.\n\nExample: let $G$ be the symmetric group on $\\set{1, \\ldots, n}$, acting on the simplex $K = \\set{x \\in \\R^n : x_i \\ge 0, \\sum x_i = 1}$ by permuting coordinates.\nThese maps are affine isometries for the Euclidean norm, and the common fixed point is the barycentre $(1/n, \\ldots, 1/n)$.\n\n## Explanations\n\n### L1 human\n\nIf a family of symmetries moves a compact convex set into itself without distorting distances, then some point is left in place by all of them at once.\nIn finite dimensions one can often take a centre of mass, but in infinite dimensions there is no obvious centre, and the theorem supplies one.\n\n### L2 human\n\nA classical application, in the locally convex form, is the existence of Haar measure on a compact group.\nThe group acts by affine maps on a compact convex set of means on the continuous functions.\nA common fixed point is an invariant mean, which gives the Haar integral.\nThe theorem should be compared with the Markov–Kakutani theorem, which asks for commuting continuous affine maps instead of isometries.\n",
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