
Journal of Convex Analysis 19 (2012), No. 3, 609630 Copyright Heldermann Verlag 2012 A Duality Principle for the Legendre Transform Goran Peskir School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, Great Britain goran@maths.man.ac.uk 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. Keywords: Legendre transform, von Neumann's minimax theorem, Fenchel's duality theorem, shortest path between graphs, obstacles, geodesic, optimal stopping problem, game, free boundary problem, Brownian motion, diffusion, Markov process, superharmonic subharm MSC: 49J35, 51M25, 60G40; 53C22, 60J65, 91A15 [ Fulltextpdf (202 KB)] for subscribers only. 