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Are there some outstanding results using some version of Helly's theorem in a totally different area (whatever that means) than convex geometry?

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    $\begingroup$ The Helly property for cat(0) cube complexes is used in geometric group theory $\endgroup$ – Benjamin Steinberg May 27 '15 at 20:03
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    $\begingroup$ Supplementing Benjamin's comment: Ivanov, Sergei. "On Helly's theorem in geodesic spaces." arXiv:1401.6654 (2014). $\endgroup$ – Joseph O'Rourke May 27 '15 at 20:16
  • $\begingroup$ Is Helly's selection theorem an infinite dimensional version of this? Same guy $\endgroup$ – john mangual May 27 '15 at 21:55
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The famous Krasnoselsky criteria for star-shaped regions is based on Helly theorem. It is important in embedding theorems for Sobolev spaces.

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Helly's theorem plays a role in economics theory and in game theory (noncooperative games):

  • Fuchs-Seliger, Susanne. "An application of Helly's theorem to preference-generated choice correspondences." International Economic Review (1984): 71-77. (Jstor link.)

  • Raghavan, T. E. S. "Zero-sum two-person games." Handbook of Game Theory 2 (1994): 735-768. (PDF download.)

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