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

*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.
A: The famous Krasnoselsky criteria for star-shaped regions is based on Helly theorem. It is important in embedding theorems for Sobolev spaces.
A: 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.)

