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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1$\begingroup$ The Helly property for cat(0) cube complexes is used in geometric group theory $\endgroup$– Benjamin SteinbergCommented May 27, 2015 at 20:03
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1$\begingroup$ Supplementing Benjamin's comment: Ivanov, Sergei. "On Helly's theorem in geodesic spaces." arXiv:1401.6654 (2014). $\endgroup$– Joseph O'RourkeCommented May 27, 2015 at 20:16
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$\begingroup$ Is Helly's selection theorem an infinite dimensional version of this? Same guy $\endgroup$– john mangualCommented May 27, 2015 at 21:55
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3 Answers
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Applications of Helly's theorem to linear programming are discussed in this thesis.
Then there are applications to the theory of approximation of continuous functions by polynomials (Chebyshev approximation).
Robotics (the discovery of an obstacle-avoiding path) makes use of Helly's theorem.
Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem discusses an application in the context of pattern recognition.
A Wavelet Approximation to the Helmholtz Equation relies on Helly's theorem.
Finally, an application to the social sciences appears in the proof of the Agreeable Society Theorem.
answered May 27, 2015 at 19:27
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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.)
answered May 27, 2015 at 20:06