Are there some outstanding results using some version of Helly's theorem in a totally different area (whatever that means) than convex geometry?
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.
The famous Krasnoselsky criteria for star-shaped 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):