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      "statement": "# Kakutani: a generalization of Brouwer's fixed-point theorem\n\nExtended Brouwer's fixed-point result to upper-semicontinuous set-valued mappings with nonempty compact convex values on a compact convex domain.\n\n## Scientific scope\n\nThe record is Kakutani's sole-author 1941 Duke Mathematical Journal paper and its finite-dimensional hypotheses. It does not absorb later infinite-dimensional extensions, Nash equilibrium or computational fixed-point algorithms.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nShizuo Kakutani\n\nThe named authors contributed to this scoped work; individual effort shares and later applications are not assigned by the byline.\n\n\n\n## Original sources\n\n- [Duke DOI for Kakutani fixed-point paper](https://doi.org/10.1215/S0012-7094-41-00838-4)\n- [UC Davis copy of Kakutani paper](https://www.math.ucdavis.edu/~deloera/TEACHING/READINGSEMINAR/PAPERS/kakutanioriginal.pdf)\n",
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