Famous theorems that are special cases of linear programming (or convex) duality

To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.

Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).

Bondareva–Shapley_theorem is quite famous in a game theory. In more modern terms, it unites the usual and dual description of the generalized permutohedron.