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An elementary direct proof that the Knaster–Kuratowski–Mazurkiewicz lemma implies Sperner’s lemma

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  • Voorneveld, Mark

Abstract

Three central results in economic theory – Brouwer’s fixed-point theorem, Sperner’s lemma, and the Knaster–Kuratowski–Mazurkiewicz (KKM) lemma – are known to be equivalent. In almost all cases, elementary direct proofs of one of these results using any of the others are easily found in the literature. This seems not to be the case for the claim that the KKM lemma implies Sperner’s lemma. The goal of this note is to provide such a proof.

Suggested Citation

  • Voorneveld, Mark, 2017. "An elementary direct proof that the Knaster–Kuratowski–Mazurkiewicz lemma implies Sperner’s lemma," Economics Letters, Elsevier, vol. 158(C), pages 1-2.
    • Handle: RePEc:eee:ecolet:v:158:y:2017:i:c:p:1-2
    DOI: 10.1016/j.econlet.2017.06.013
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    Cited by:

    1. Petri, Henrik & Voorneveld, Mark, 2016. "No bullying! A playful proof of Brouwer's fixed-point theorem," SSE Working Paper Series in Economics 2016:3, Stockholm School of Economics, revised 20 Jun 2017.
    2. Petri, Henrik & Voorneveld, Mark, 2018. "No bullying! A playful proof of Brouwer’s fixed-point theorem," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 1-5.

    More about this item

    Keywords

    Knaster–Kuratowski–Mazurkiewicz; Sperner; Brouwer;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other

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