Isometries for the Legendre-Fenchel transform

It is shown that on the space of lower semicontinuous convex functions defined on R n {R^n} , the conjugation map—the Legendre-Fenchel transform—is an isometry with respect to some metrics consistent with the epi-topology. We also obtain isometries for the infinite dimensional case (Hilbert space and reflexive Banach space), but this time they correspond to topologies finer than the Moscoepi-topology.