Content:
It is shown that on the space of lower semicontinuous convex functions defined on the n-dimensional Euclidean space, 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 Mosco-epi-topology.
January 2000