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- W4288602207 abstract "In 1967 Herbert Scarf suggested a new proof of Brouwer's fixed point theorem based on a combinatorial analogue of Sperner's lemma. Scarf presented his arguments in very geometric language, even purely combinatorial ones. Recently H. Petri and M. Voorneveld published an almost geometry-free version of Scarf's proof. Their version eliminated even only implicitly geometric aspects of Scarf's proof, namely, the structure of an abstract simplicial complex behind the combinatorial arguments. The present paper is devoted to a proof of Scarf's analogue of Sperner's lemma in the abstract setting of a collection of linear orders on a finite set. This proof partially follows the proof by Petri and Voorneveld, but restores the implicit geometry to its rightful place. We also deduce Brouwer's fixed point theorem from this analogue and discuss various versions of Scarf's proof." @default.
- W4288602207 created "2022-07-29" @default.
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- W4288602207 date "2019-02-02" @default.
- W4288602207 modified "2023-10-18" @default.
- W4288602207 title "Beyond Sperner's lemma" @default.
- W4288602207 doi "https://doi.org/10.48550/arxiv.1902.00827" @default.
- W4288602207 hasPublicationYear "2019" @default.
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