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The Extreme Points of Fusions

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Listed:
  • Andreas Kleiner
  • Benny Moldovanu
  • Philipp Strack
  • Mark Whitmeyer

Abstract

Our work explores fusions, the multidimensional counterparts of mean-preserving contractions and their extreme and exposed points. We reveal an elegant geometric/combinatorial structure for these objects. Of particular note is the connection between Lipschitz-exposed points (measures that are unique optimizers of Lipschitz-continuous objectives) and power diagrams, which are divisions of a space into convex polyhedral ``cells'' according to a weighted proximity criterion. These objects are frequently seen in nature--in cell structures in biological systems, crystal and plant growth patterns, and territorial division in animal habitats--and, as we show, provide the essential structure of Lipschitz-exposed fusions. We apply our results to several questions concerning categorization.

Suggested Citation

  • Andreas Kleiner & Benny Moldovanu & Philipp Strack & Mark Whitmeyer, 2024. "The Extreme Points of Fusions," Papers 2409.10779, arXiv.org.
    • Handle: RePEc:arx:papers:2409.10779
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    References listed on IDEAS

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    1. Piotr Dworczak & Giorgio Martini, 2019. "The Simple Economics of Optimal Persuasion," Journal of Political Economy, University of Chicago Press, vol. 127(5), pages 1993-2048.
    2. Andreas Kleiner & Benny Moldovanu & Philipp Strack, 2021. "Extreme Points and Majorization: Economic Applications," Econometrica, Econometric Society, vol. 89(4), pages 1557-1593, July.
    3. Maxim Ivanov, 2021. "Optimal monotone signals in Bayesian persuasion mechanisms," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 72(3), pages 955-1000, October.
    4. Gerhard Winkler, 1988. "Extreme Points of Moment Sets," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 581-587, November.
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