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  "body": "\n## Statement\n\nHidetoshi Komiya generalized the KKMS theorem from simplices to polytopes. Let *P* be any compact convex polytope. Let $\\textrm{Faces}(P)$ be the set of nonempty faces of *P*. A **Komiya covering** of *P* is a family of closed sets $\\{C_F: F\\in \\textrm{Faces}(P)\\}$ such that for every face $F\\in \\textrm{Faces}(P)$: $F\\subseteq \\bigcup_{G\\subseteq F, ~ G \\in \\textrm{Faces}(P)} C_G.$ Komiya's theorem says that for every Komiya covering of *P*, there is a *balanced collection* $B\\subseteq \\textrm{Faces}(P)$, such that the intersection of sets indexed by $B$ is nonempty:\n\n$$\\bigcap_{F\\in B} C_F \\neq \\emptyset$$\n\nKomiya's theorem also generalizes the definition of a balanced collection: instead of requiring that there is a weight function on $B$ such that the sum of weights near each vertex of *P* is 1, we start by choosing any set of points $\\textbf{b} = \\{b^F: F\\in \\textrm{Faces}(P), b^F\\in F \\}$. A collection $B\\subseteq \\textrm{Faces}(P)$ is called *balanced* with respect to $\\textbf{b}$ iff $b^P \\in \\operatorname{conv} \\{ b^F: F\\in B \\}$, that is, the point assigned to the entire polygon *P* is a convex combination of the points assigned to the faces in the collection *B*.\n\nThe KKMS theorem is a special case of Komiya's theorem in which the polytope $P = \\Delta_{n-1}$ and $b^F$ is the barycenter of the face F (in particular, $b^P$ is the barycenter of $\\Delta_{n-1}$, which is the point $(1/n, \\ldots, 1/n)$ ).\n",
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        "locator": "Knaster–Kuratowski–Mazurkiewicz lemma, section Generalizations \u003e Polytopal KKMS theorem (Komiya)",
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