A Duality Principle for the Legendre Transform

We present a duality principle for the Legendre transform that yields the shortest path between the graphs of functions and embodies the underlying Nash equilibrium. A useful feature of the algorithm for the shortest path obtained in this way is that its implementation has a local character in the sense that it is applicable at any point in the domain with no reference to calculations made earlier or elsewhere. The derived results are applied to optimal stopping games of Brownian motion and diffusion processes where the duality principle corresponds to the semiharmonic characterisation of the value function.