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

1$\begingroup$ The Helly property for cat(0) cube complexes is used in geometric group theory $\endgroup$ – Benjamin Steinberg May 27 '15 at 20:03

1$\begingroup$ 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
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 obstacleavoiding path) makes use of Helly's theorem.
Analysis of Incomplete Data and an IntrinsicDimension 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.
The famous Krasnoselsky criteria for starshaped regions is based on Helly theorem. It is important in embedding theorems for Sobolev spaces.
Helly's theorem plays a role in economics theory and in game theory (noncooperative games):
FuchsSeliger, Susanne. "An application of Helly's theorem to preferencegenerated choice correspondences." International Economic Review (1984): 7177. (Jstor link.)
Raghavan, T. E. S. "Zerosum twoperson games." Handbook of Game Theory 2 (1994): 735768. (PDF download.)